Purplemath What is a ratio? A "ratio" is just a comparison between, or a relating of, two different things. Ratios are used to create proportions by setting two ratios equal to each other and solving for some unknown, and ratios can also be used to find per-unit rates such as how many mile a car can drive "per liter" or how many hours the average student at a …To factor a quadratic (that is, to factor a trinomial of the form ax2 + bx + c) where the leading coefficient a is not equal to 1, follow these steps: Multiply the leading coefficient a and the constant term c to get the product ac. Find factors of ac that add up to the coefficient of the constant term b. Use these factors of ac to split the ... Purplemath What is an identity? In mathematics, an "identity" is an equation which is always true, regardless of the specific value of a given variable. An identity can be "trivially" true, such as the equation x = x or an identity can be usefully true, such as the Pythagorean Theorem's a 2 + b 2 = c 2 So x = 1 is one of the zeroes. Trying x = −1, I get: 1 − 9 + 11 + 22 − 9 + 11 + 21 = 48. Okay; so that one isn't a zero. But, to reduce my polynomial by the one factor corresponding to this zero, I'll do my first synthetic division: So my reduced polynomial is equation is: x5 + 10 x4 + 21 x3 − x2 − 10 x − 21 = 0. Now I can solve each factor by setting each one equal to zero and solving the resulting linear equations: x + 2 = 0 or x + 3 = 0. x = −2 or x = − 3. These two values are the solution to the original quadratic equation. So my answer is: x = −3, −2.Purplemath What are exponents (in math)? Exponents, also called powers or orders, are shorthand for repeated multiplication of the same thing by itself. For instance, the shorthand for multiplying three copies of the number 5 is shown on the right-hand side of the "equals" sign in (5)(5)(5) = 5 3.This proportionality of corresponding sides can be used to find the length of a side of a figure, given a similar figure for which sufficient measurements are known. In the displayed triangles, the lengths of the sides are given by A = 48 mm, B = 81 mm, C = 68 mm, and a = 21 mm. Find the lengths of sides b and c, rounded to the nearest …Purplemath. An important category of percentage exercises is markup and markdown problems. For these, you calculate the markup or markdown of the price or cost in absolute terms (you find by how much the price or cost changed), and then you calculate the percent change relative to the original value. So they're really …To graph a log function: Always keep in mind that logs are inverses of exponentials; this will remind you of the shape you should expect the graph to have. Pick input values (that is, x -values) that are powers of the base; for instance, if the log's base is 5, then pick x -values like 52 and 5−1. List the corresponding y -values; for ...Use completing the square to solve x2 − 4x − 8 = 0. As noted above, this quadratic does not factor, so I can't solve the equation by factoring. And they haven't given me the equation in a form that is ready to square-root. But there is a way for me to manipulate the quadratic to put it into that ready-for-square-rooting form, so I can solve.Purplemath. Sometimes functions need to have their domains restricted, in order for the function to be invertible. On the other hand, some functions come with their own domain restrictions. Rational functions, for example, have variables in their denominators, and their domains may therefore be restricted, in order to avoid … The Purplemath algebra lessons are available in offline form for home use! This allows you to, for instance, review the lessons on your laptop while you ride the bus, or let your grandkids "surf" the site without having to provide them with a "live" Internet connection. The "Purplemath CD" contains the entire Purplemath web site, modified for ... So x = 1 is one of the zeroes. Trying x = −1, I get: 1 − 9 + 11 + 22 − 9 + 11 + 21 = 48. Okay; so that one isn't a zero. But, to reduce my polynomial by the one factor corresponding to this zero, I'll do my first synthetic division: So my reduced polynomial is equation is: x5 + 10 x4 + 21 x3 − x2 − 10 x − 21 = 0. Purplemath. While slogging through these exercises, you may have wondered: How does partial fraction decomposition work? Partial fraction decomposition works by using prime factors and some logic to take apart complicated fractions into smaller, simpler ones. Content Continues Below.Purplemath. Sometimes functions need to have their domains restricted, in order for the function to be invertible. On the other hand, some functions come with their own domain restrictions. Rational functions, for example, have variables in their denominators, and their domains may therefore be restricted, in order to avoid …Improve your SAT math score with online test prep classes from PurpleMath and MathHelp. Free SAT practice questions and a personal math tutor!To factor a quadratic (that is, to factor a trinomial of the form ax2 + bx + c) where the leading coefficient a is not equal to 1, follow these steps: Multiply the leading coefficient a and the constant term c to get the product ac. Find factors of ac that add up to the coefficient of the constant term b. Use these factors of ac to split the ...Purplemath. Back when you first studied square roots and how to solve radical equations, you were probably introduced to something called "the Pythagorean Theorem". This Theorem relates the lengths of the three sides of any right triangle. This Theorem existed way before Pythagorus and his followers, the …Purplemath. Another "typical" work problem is the "one guy did part of the job" or "the number of workers changed at some point during the job" type. We'll still need to do the computations for how much each guy does per unit time (usually hours or days), but we may need to use the fact that "a completed task" is represented by " …can be written as 0.538461538461…. These two fractions are repeating decimals. In the first case, the repeated block is just 3; in the second case, the repeated block is 538461.. On the other hand, we have loads of other numbers whose decimal forms are non-repeating, non-terminating decimals; these number are non-rational (that is, they cannot be written as …Purplemath. In the previous two pages, we've looked at solving one-step linear equations; that is, equations that require one addition or subtraction, or that require one multiplication or division. However, most linear equations require more than one step in order to find their solution. What steps then should be used, and in what order?y ≥ (2/3) x − 4. y ≤ (−1/5) x + 4. x > 0. "Solving" systems of two-variable linear inequalities means "graphing each individual inequality, and then finding the overlaps of the various solutions". So I graph each inequality individually, marking the "solution" side of each line as I go, and then I'll find the overlapping portion of the ... 2. 1. 0. The first row above (labelled "digits") contains the digits from the binary number; the second row (labelled "numbering") contains the power of 2 (the base) corresponding to each digit. I will use this listing to convert each digit to the power of two that it represents: 1×2 8 + 0×2 7 + 1×2 6 + 1×2 5 + 0×2 4 + 0×2 3 + 1×2 2 + 0 ... Purplemath. I've listed many logs rules, and so far we've used all but the Change-of-Base Formula. (Okay, we haven't used the Base-Switch Rule, but I don't know where that would be useful anyway, …Purplemath. Up until now, you've been told that you can't take the square root of a negative number. That's because you had no numbers which were negative after you'd squared them — so you couldn't "go backwards" and return to them by taking the square root. Before now, every number was positive after you squared it.Purplemath. Variation problems aren't hard once you get the hang of the lingo. The only real difficulty is learning the somewhat specialized vocabulary and the techniques for this …The Binomial Theorem is a quick way (okay, it's a less slow way) of expanding (that is, of multiplying out) a binomial expression that has been raised to some (generally inconveniently large) power. For instance, the expression (3x − 2) is a binomial, 10 is a rather large exponent, and (3x − 2)10 would be very painful to multiply out by hand.Purplemath Linear programming is the process of taking various linear inequalities (called "constraints") relating to some situation, and finding the best value obtainable under those conditions. A typical example would be taking the limitations of materials and labor, and then determining the optimal production levels for maximal profits …Find the mean, median, mode, and range for the following list of values: 1, 2, 4, 7. The mean is the usual average: (1 + 2 + 4 + 7) ÷ 4 = 14 ÷ 4 = 3.5. The median is the middle number. In this example, the numbers are already listed in numerical order, so I don't have to rewrite the list. But there is no "middle" number, because there are …You can solve this "space" problem by using negative numbers. The "whole" numbers start at zero and count off to the right; these are the positive integers. The negative integers start at zero and count off to the left: Note the arrowhead on the far right end of the number line above. That arrow tells you the direction in which the … The Purplemath lessons have been written so that they may be studied in whatever manner the student finds most useful. Different textbooks cover different topics in different orders. The Purplemath lessons try not to assume any fixed ordering of topics, so that any student, regardless of the textbook being, may benefit. You should know the formula for the circumference C and area A of a circle, given the radius r: Acir = π r2. Ccir = 2π r. (" π " is the number approximated by 3.14159 or the fraction 22/7) Remember that the radius of a circle is the distance from the center to the outside of a circle. In other words, the radius is just halfway across.The distinction between the two formulas is in the location of that one "minus" sign: For the difference of cubes, the "minus" sign goes in the linear factor, a − b; for the sum of cubes, the "minus" sign goes in the quadratic factor, a2 − ab + b2. Some people use the mnemonic " SOAP " to help keep track of the signs; the letters …Spend time reading and practice your writing skills. Make use of a TSI math practice test to defeat any word problem anxiety. Improve your tactics for good test taking. Study until you feel certain of your abilities. Improve your TSI math score with online test prep classes from PurpleMath and MathHelp.To factor a quadratic (that is, to factor a trinomial of the form ax2+ bx+ c) where the leading coefficient a is not equal to 1, follow these steps: Multiply the leading coefficient a and the constant term c to get the product ac. Find factors of ac that add up to the coefficient of the constant term b. Use these factors of ac to split the ...Purplemath. Venn diagram word problems generally give you two or three classifications and a bunch of numbers. You then have to use the given information to populate the diagram and figure out the remaining information. For instance: Out of forty students, 14 are taking English Composition and 29 are taking Chemistry.Purplemath What is a circle? A circle is a geometrical shape. It is defined as having a center, and being the set of all points that are a certain fixed distance from that center. (The fixed distance is called the radius of the circle.) The circle is not of much use in algebra since the equation of a circle isn't a function. So x = 1 is one of the zeroes. Trying x = −1, I get: 1 − 9 + 11 + 22 − 9 + 11 + 21 = 48. Okay; so that one isn't a zero. But, to reduce my polynomial by the one factor corresponding to this zero, I'll do my first synthetic division: So my reduced polynomial is equation is: x5 + 10 x4 + 21 x3 − x2 − 10 x − 21 = 0. MathHelp.com. Step 1 in effectively translating and solving word problems is to read the problem entirely. Don't start trying to solve anything when you've only read half a sentence. Try first to get a feel for the whole problem; try first to see what information you have, and then figure out what you still need. Advertisement. The Rational Roots Test (or Rational Zeroes Theorem) is a handy way of obtaining a list of useful first guesses when you are trying to find the zeroes (or roots) of a polynomial. Given a polynomial with integer (that is, positive and negative whole-number) coefficients, the *possible* zeroes are found by listing the …Purplemath. An arithmetic series is the sum of the terms of an arithmetic sequence. A geometric series is the sum of the terms of a geometric sequence. There are other types of series, but you're unlikely to work with them much until you're in calculus. For now, you'll probably mostly work with these two. This page explains and illustrates …For the three-sevenths fraction, the denominator needed a factor of 5, so I multiplied by \frac {5} {5} 55, which is just 1. We can use this same technique to rationalize radical denominators. I could take a 3 out of the denominator of my radical fraction if I had two factors of 3 inside the radical.What are other number bases called? We use the decimal number base, having ten digits; other number bases have their own names. For instance, the base-11 number base is called the "undecimal" base; base-12 is called "dozenal" (as in, "it has a dozen digits").The base-8 system is called "octal"; the base-16 system is called "hexidecimal"; the base-2 system …Purplemath. Unlike the examples on the previous page, nearly all polynomial divisions do not "come out even"; usually, you'll end up with a remainder. Divide 3x 3 − 5x 2 + 10x − 3 by 3x + 1; I start with the long-division set-up: Looking only at the leading terms, I divide 3x 3 by 3x to get x 2. This is what I put on top:Purplemath. Once you've learned the basic keywords for translating word problems from English into mathematical expressions and equations, you'll be presented with various English expressions, and be told to perform the translation. Don't view the lists of keywords as holy writ, handed down from on high. Instead, use these lists …Purplemath What is an angle of elevation / inclination? An angle of elevation (also called an angle of inclination) is an angle that goes above the horizontal from whatever is the vantage point. For instance, suppose you are standing on the sidewalk looking up at the top of the chimney on the house across the street.Learn how to find real-number solutions and factors of polynomials using synthetic division, rational roots test, and quadratic formula. See detailed steps and graphs for each … Solve x2 − 48 = 0. This quadratic expression has two terms, and nothing factors out, so either it's a difference of squares (which I can factor) or else it can be formatted as " (variable part) 2 equals (a number)" so I can square-root both sides. Since 48 is not a square, I can't apply the difference-of-squares formula. Purplemath. Most exponential equations do not solve neatly; there will be no way to convert the bases to being the same, such as the conversion of 4 and 8 into powers of 2. In solving these more-complicated equations, you will have to use logarithms. So x = 1 is one of the zeroes. Trying x = −1, I get: 1 − 9 + 11 + 22 − 9 + 11 + 21 = 48. Okay; so that one isn't a zero. But, to reduce my polynomial by the one factor corresponding to this zero, I'll do my first synthetic division: So my reduced polynomial is equation is: x5 + 10 x4 + 21 x3 − x2 − 10 x − 21 = 0. Purplemath. Variation problems aren't hard once you get the hang of the lingo. The only real difficulty is learning the somewhat specialized vocabulary and the techniques for this classification of problems. Variation problems involve fairly simple relationships or formulas, involving one variable being equal to one term.The general form of a parabola's equation is the quadratic that you're used to: y = ax2 + bx + c. — unless the quadratic is sideways, in which case the equation will look something like this: x = ay2 + by + c. The important difference in the two equations is in which variable is squared: for regular (that is, for vertical) parabolas, the x ...Purplemath What are exponents (in math)? Exponents, also called powers or orders, are shorthand for repeated multiplication of the same thing by itself. For instance, the shorthand for multiplying three copies of the number 5 is shown on the right-hand side of the "equals" sign in (5)(5)(5) = 5 3.You should know the formula for the circumference C and area A of a circle, given the radius r: Acir = π r2. Ccir = 2π r. (" π " is the number approximated by 3.14159 or the fraction 22/7) Remember that the radius of a circle is the distance from the center to the outside of a circle. In other words, the radius is just halfway across.To factor a quadratic (that is, to factor a trinomial of the form ax2 + bx + c) where the leading coefficient a is not equal to 1, follow these steps: Multiply the leading coefficient a and the constant term c to get the product ac. Find factors of ac that add up to the coefficient of the constant term b. Use these factors of ac to split the ... Purplemath What is a circle? A circle is a geometrical shape. It is defined as having a center, and being the set of all points that are a certain fixed distance from that center. (The fixed distance is called the radius of the circle.) The circle is not of much use in algebra since the equation of a circle isn't a function. Purplemath. In addition to finding lines (axes) of symmetry, you can also look for points of symmetry. A point of symmetry is a point that represents a "center" of sorts for the figure. For any line that you draw through the point of symmetry, if this line crosses the figure on one side of the point, the line will also cross the figure on the ...Purplemath What is engineering notation? Engineering notation is similar to scientific notation, in that numbers are converted to (a number) times (10 raised to some power). But the powers in engineering notation will always be multiples of 3.. Because the powers are always multiples of three, the resulting numbers …The foci are side by side, so this hyperbola's branches are side by side, and the center, foci, and vertices lie on a line paralleling the x -axis. So the y part of the equation will be subtracted and the a2 will go with the x part of the equation. The center is midway between the two foci, so the center must be at (h, k) = (−1, 0).The Algebra 1 course, often taught in the 9th grade, covers Linear equations, inequalities, functions, and graphs; Systems of equations and inequalities; Extension of the concept of a function; Exponential models; and Quadratic equations, functions, and graphs. Khan Academy's Algebra 1 course is built to deliver a comprehensive, illuminating, engaging, …y ≥ (2/3) x − 4. y ≤ (−1/5) x + 4. x > 0. "Solving" systems of two-variable linear inequalities means "graphing each individual inequality, and then finding the overlaps of the various solutions". So I graph each inequality individually, marking the "solution" side of each line as I go, and then I'll find the overlapping portion of the ...Purplemath is a website that provides free math lessons and resources for students and teachers. It started in 1998 as a personal web site by Elizabeth Stapel, and has grown to …Describe the end behavior of f (x) = 3x7 + 5x + 1004. This polynomial is much too large for me to view in the standard screen on my graphing calculator, so either I can waste a lot of time fiddling with WINDOW options, or I can quickly use my knowledge of end behavior. This function is an odd-degree polynomial, so the ends go off in opposite ...Purplemath. The "addition" method of solving systems of linear equations is also called the "elimination" method. Under either name, this method is similar to the method you probably used when you were first learning how to solve one-variable linear equations. Suppose, back in the day, they'd given you the equation " x + 6 = 11 ".Purplemath. Back when you first studied square roots and how to solve radical equations, you were probably introduced to something called "the Pythagorean Theorem". This Theorem relates the lengths of the three sides of any right triangle. This Theorem existed way before Pythagorus and his followers, the …To multiply a matrix by a scalar, multiply each entry of the matrix by the scalar's value. For instance, given a matrix M and the scalar −1, the scalar product −1M will multiply each entry in M by −1, so each entry in −1M will have the opposite sign of each entry in the original matrix M.Purplemath. Venn diagrams were invented by a guy named John Venn (no kidding; that was really his name) as a way of picturing relationships between different groups of things. Inventing this type of diagram was, apparently, pretty much all John Venn ever accomplished. To add insult to injury, much of what we refer to as "Venn … Using these numbers, I can split the middle −13x term into the two terms −9x and −4x, and then I can factor in pairs: 6 x2 − 13x + 6. = 6 x2 − 9x − 4x + 6. = 3 x (2 x − 3) − 2 (2 x − 3) = (2x − 3) (3x − 2) The factoring method in the last two examples above — in particular, the part where I picked two numbers for ... The Purple Comet! Math Meet needs your small voluntary contribution to survive. See complete problem solutions 2003-2012 with the first Purple Comet Book and …To find the selling price per pound of the mixture, divide ( $139.60) by ( 20 pounds). Simplify the division to find the unit rate. Remember to put appropriate units (in this case, "dollars per pound") on your hand-in answer. Note that, in this case, no variable was actually necessary.Purplemath. The first type of logarithmic equation has two logs, each having the same base, which have been set equal to each other. We solve this sort of equation by setting the insides (that is, setting the "arguments") of the logarithmic expressions equal to each other. For example: Solve log 2 (x) = log 2 (14).Purplemath What is a fraction? A fraction is a ratio of two whole numbers, such as ¾. The number on top is called the numerator; the number underneath is called the denominator. The word numerator is derived from a Latin word meaning "counter"; the word denominator is derived from a Latin word meaning "name".Note this common technique: In the "n = k + 1" step, it is usually a good first step to write out the whole formula in terms of k + 1, and then break off the "n = k", so you can replace it with whatever assumption you made about n = k in the assumption step.Then you manipulate and simplify, and try to rearrange things to get the RHS … Purplemath. When you're graphing (or looking at a graph of) polynomials, it can help to already have an idea of what basic polynomial shapes look like. One of the aspects of this is "end behavior", and it's pretty easy. We'll look at some graphs, to find similarities and differences. First, let's look at some polynomials of even degree ... Purplemath, Addison, Illinois. 3.3K likes · 82 talking about this. https://www.purplemath.com Need help with algebra? Try Purplemath's practical and …1 foot : 12 inches. 2.54 centimeters : 1 inch. 100 centimeters : 1 meter. I could have chosen other conversion factors, if I'd felt like it. But these factors provide connections, one way or another, between "seconds" and "hours" and between "miles" and "meters", so they'll get the job done. Content Continues Below. In sum, the steps for graphing radical (that is, square root) functions are these: Find the domain of the function: set the insides of the radical "greater than or equal to" zero, and solve for the allowable x -values. Make a T-chart to hold your plot points. Pick x -values within the domain (including the "or equal to" endpoint of the domain ... Purplemath. Back when you first studied square roots and how to solve radical equations, you were probably introduced to something called "the Pythagorean Theorem". This Theorem relates the lengths of the three sides of any right triangle. This Theorem existed way before Pythagorus and his followers, the … Evaluate 6!. A factorial is just a product. To "evaluate" a factorial is simply to multiply it out. In this case, they're wanting me to "take the factorial of" 6. This means that I need to multiply all the whole numbers from 1 through 6, inclusive. My work is pretty simple: 1×2×3×4×5×6 = 720. This value is all they're looking for, so my ... Solve (x + 1) (x − 3) = 0. To solve this quadratic equation, I could multiply out the expression on the left-hand side, simplify to find the coefficients, plug those coefficient values into the …Purplemath What are a number's "factors"? "Factors" are the whole numbers you multiply to get another whole number. For instance, factors of 15 are 3 and 5, because 3 × 5 = 15. Some numbers have more than one factorization (more than one way of being factored). For instance, 12 can be factored as 1 ×12, 2 × 6, and also as 3 × 4.Learn how to find real-number solutions and factors of polynomials using synthetic division, rational roots test, and quadratic formula. See detailed steps and graphs for each …The first solution is 45° more than a multiple of 180°, so (180n)° + 45° should do. The second solution is 30° more than a multiple of 180° and (because of the "plus / minus") also 30° less than that same multiple, so (180n)° ± 30° will cover this part. x = (180n)° ± 30°, (180n)° + 45° for all integers n. 2. 1. 0. The first row above (labelled "digits") contains the digits from the binary number; the second row (labelled "numbering") contains the power of 2 (the base) corresponding to each digit. I will use this listing to convert each digit to the power of two that it represents: 1×2 8 + 0×2 7 + 1×2 6 + 1×2 5 + 0×2 4 + 0×2 3 + 1×2 2 + 0 ... Thomas automotive, The ring boxes, Downriver medical associates, Nolas tulsa, Powerhouse motorsports, Camdenton missouri, Rocky mountain cancer centers, King pins, Steele gmc round rock, Nbc soccer, Kcal rocks, Heb mission tx, Secretary of state california, Comfort inn sturbridge ma
Solve (x + 1) (x − 3) = 0. To solve this quadratic equation, I could multiply out the expression on the left-hand side, simplify to find the coefficients, plug those coefficient values into the …. Wrestling tracker
e.c. nurrePurplemath What is a fraction? A fraction is a ratio of two whole numbers, such as ¾. The number on top is called the numerator; the number underneath is called the denominator. The word numerator is derived from a Latin word meaning "counter"; the word denominator is derived from a Latin word meaning "name". Purplemath. Most exponential equations do not solve neatly; there will be no way to convert the bases to being the same, such as the conversion of 4 and 8 into powers of 2. In solving these more-complicated equations, you will have to use logarithms. Free math problem solver answers your algebra homework questions with step-by-step explanations. 3.141 | 59265... The number in the fourth place is a 5, which is the cut-off for rounding: if the number in the next place (after the one you're rounding to) is 5 or greater, you round up. In this case, the 1 becomes a 2, the 59265... part disappears, and π, rounded to three decimal places, is: 3.142. Content Continues Below.The four directions in which one can move a function's graph are up, down, to the right, and to the left. Usually, translation involves only moving the graph around. Squeezing or stretching a graph is more of a "transformation" of the graph. But these two topics are usually taught at the same time, and usually under the same name.The basic metric units are meters (for length), grams (for mass or weight), and liters (for volume). And the different units convert into one another rather nicely, with one milliliter equalling one cubic centimeter (where one Cubic Centimeter is the "cc" of medical shows on television) and one gram being the mass (or weight) of one cc …Purplemath. Variation problems aren't hard once you get the hang of the lingo. The only real difficulty is learning the somewhat specialized vocabulary and the techniques for this classification of problems. Variation problems involve fairly simple relationships or formulas, involving one variable being equal to one term.The two rules for function reflection are these: To reflect the graph of a function h(x) over the x -axis (that is, to flip the graph upside-down), multiply the function by −1 to get −h(x). To reflect the graph of a function h(x) around the y -axis (that is, to mirror the two halves of the graph), multiply the argument of the function by ...Also, this hyperbola's foci and vertices are to the left and right of the center, on a horizontal line paralleling the x -axis. From the equation, clearly the center is at (h, k) = (−3, 2). Since the vertices are a = 4 units to either side, then they are at the points (−7, 2) and at (1, 2). The equation a2 + b2 = c2 gives me:Find the mean, median, mode, and range for the following list of values: 1, 2, 4, 7. The mean is the usual average: (1 + 2 + 4 + 7) ÷ 4 = 14 ÷ 4 = 3.5. The median is the middle number. In this example, the numbers are already listed in numerical order, so I don't have to rewrite the list. But there is no "middle" number, because there are …The Purple Comet! Math Meet needs your small voluntary contribution to survive. See complete problem solutions 2003-2012 with the first Purple Comet Book and … Purplemath. In the previous two pages, we've looked at solving one-step linear equations; that is, equations that require one addition or subtraction, or that require one multiplication or division. However, most linear equations require more than one step in order to find their solution. What steps then should be used, and in what order? So x = 1 is one of the zeroes. Trying x = −1, I get: 1 − 9 + 11 + 22 − 9 + 11 + 21 = 48. Okay; so that one isn't a zero. But, to reduce my polynomial by the one factor corresponding to this zero, I'll do my first synthetic division: So my reduced polynomial is equation is: x5 + 10 x4 + 21 x3 − x2 − 10 x − 21 = 0. Purplemath. A ratio is one thing or value compared with or related to another thing or value; it is just a statement or an expression, and can only perhaps be simplified or reduced. On the other hand, a proportion is two ratios which have been set equal to each other; a proportion is an equation that can be solved. What are other number bases called? We use the decimal number base, having ten digits; other number bases have their own names. For instance, the base-11 number base is called the "undecimal" base; base-12 is called "dozenal" (as in, "it has a dozen digits").The base-8 system is called "octal"; the base-16 system is called "hexidecimal"; the base-2 system …Purplemath. Most exponential equations do not solve neatly; there will be no way to convert the bases to being the same, such as the conversion of 4 and 8 into powers of 2. In solving these more-complicated equations, you will have to use logarithms.To factor a quadratic (that is, to factor a trinomial of the form ax2 + bx + c) where the leading coefficient a is not equal to 1, follow these steps: Multiply the leading coefficient a and the constant term c to get the product ac. Find factors of ac that add up to the coefficient of the constant term b. Use these factors of ac to split the ...Purplemath. An important category of percentage exercises is markup and markdown problems. For these, you calculate the markup or markdown of the price or cost in absolute terms (you find by how much the price or cost changed), and then you calculate the percent change relative to the original value. So they're really …Free math problem solver answers your algebra homework questions with step-by-step explanations.Sequences and series are most useful when there is a formula for their terms. For instance, if the formula for the terms a n of a sequence is defined as "a n = 2n + 3", then you can find the value of any term by plugging the value of n into the formula. For instance, a 8 = 2(8) + 3 = 16 + 3 = 19.In words, "a n = 2n + 3" can be read as … Purplemath What is a circle? A circle is a geometrical shape. It is defined as having a center, and being the set of all points that are a certain fixed distance from that center. (The fixed distance is called the radius of the circle.) The circle is not of much use in algebra since the equation of a circle isn't a function. Purplemath. In the previous two pages, we've looked at solving one-step linear equations; that is, equations that require one addition or subtraction, or that require one multiplication or division. However, most linear equations require more than one step in order to find their solution. What steps then should be used, and in what order? Find the mean, median, mode, and range for the following list of values: 1, 2, 4, 7. The mean is the usual average: (1 + 2 + 4 + 7) ÷ 4 = 14 ÷ 4 = 3.5. The median is the middle number. In this example, the numbers are already listed in numerical order, so I don't have to rewrite the list. But there is no "middle" number, because there are …Note this common technique: In the "n = k + 1" step, it is usually a good first step to write out the whole formula in terms of k + 1, and then break off the "n = k", so you can replace it with whatever assumption you made about n = k in the assumption step.Then you manipulate and simplify, and try to rearrange things to get the RHS …Homework Guidelines for Mathematics. Mathematics is a language, and as such it has standards of writing which should be observed. In a writing class, one must respect the …Purplemath What is synthetic division? Synthetic division is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor — and it only works in this case. Synthetic division is generally used, however, not for dividing out factors but for finding zeroes (or roots) of polynomials.Homework Guidelines for Mathematics. Mathematics is a language, and as such it has standards of writing which should be observed. In a writing class, one must respect the …Purplemath. Even when studying algebra, one sometimes needs notation from other areas, such as geometry. After algebra, one usually studies trigonometry and then calculus. Content Continues Below. MathHelp.com. The following table includes geometric, trigonometric, probability, and aditional mathematical notation.The Binomial Theorem is a quick way (okay, it's a less slow way) of expanding (that is, of multiplying out) a binomial expression that has been raised to some (generally inconveniently large) power. For instance, the expression (3x − 2) is a binomial, 10 is a rather large exponent, and (3x − 2)10 would be very painful to multiply out by hand.Purplemath. Graphing exponential functions is similar to the graphing you have done before. However, by the nature of exponential functions, their points tend either to be very close to one fixed value or else to be too large to be conveniently graphed. In fact, there will generally be only a few points that are reasonable to use for … Purplemath What are exponents (in math)? Exponents, also called powers or orders, are shorthand for repeated multiplication of the same thing by itself. For instance, the shorthand for multiplying three copies of the number 5 is shown on the right-hand side of the "equals" sign in (5)(5)(5) = 5 3. Logarithms are inverse functions (backwards), and logs represent exponents (concept), and taking logs is the undoing of exponentials (backwards and a concept). And this is a lot to take in all at once. Yes, in a sense, logarithms are themselves exponents. Logarithms have bases, just as do exponentials; for instance, log5(25) …In an intuitive sense, the Midpoint Formula takes the coordinates of the two given points, and finds the averages of the x - and y -values. Think about it this way: If you are given two numbers, you can find the number exactly midway between them by averaging them; that is, by adding them together and dividing their sum by 2.Compound (or compounded) interest is interest that is earned on interest. If you invest $300 in a compound-interest fund for two years at 10% interest annually, you will earn $30 for the first year, but then you will earn 10% of $330 (or $33) for the second year, for a total of $63 in interest. Content Continues Below. Purplemath. In the previous two pages, we've looked at solving one-step linear equations; that is, equations that require one addition or subtraction, or that require one multiplication or division. However, most linear equations require more than one step in order to find their solution. What steps then should be used, and in what order? When you see that you have a two-term non-linear polynomial, check to see if it fits any of the formulas. In this case, you've got a difference of squares, so apply that formula: 2x2 − 162 = 2 (x2 − 81) = 2 (x − 9) (x + 9). Warning: Always remember that, in cases like 2x2 + 162, all you can do is factor out the 2; the sum of squares …Evaluate 6!. A factorial is just a product. To "evaluate" a factorial is simply to multiply it out. In this case, they're wanting me to "take the factorial of" 6. This means that I need to multiply all the whole numbers from 1 through 6, inclusive. My work is pretty simple: 1×2×3×4×5×6 = 720. This value is all they're looking for, so my ...The four directions in which one can move a function's graph are up, down, to the right, and to the left. Usually, translation involves only moving the graph around. Squeezing or stretching a graph is more of a "transformation" of the graph. But these two topics are usually taught at the same time, and usually under the same name.Classify the following equations according to the type of conic each represents: A) 3 x2 + 3 y2 − 6 x + 9 y − 14 = 0. B) 6 x2 + 12 x − y + 15 = 0. C) x2 + 2 y2 + 4 x + 2 y − 27 = 0. D) x2 − y2 + 3 x − 2 y − 43 = 0. A) Both variables are squared, and both squared terms are multiplied by the same number, so this is a circle.Logarithms are inverse functions (backwards), and logs represent exponents (concept), and taking logs is the undoing of exponentials (backwards and a concept). And this is a lot to take in all at once. Yes, in a sense, logarithms are themselves exponents. Logarithms have bases, just as do exponentials; for instance, log5(25) … The solving process works like this: 2 y − 4 x = 3. 2 y = 4 x + 3. y = 2 x + 1.5. Then we can graph as usual. By the way, it's often a good idea to use x -values which are spread out a bit. If the plotted points are too close together, we can end up not being quite sure of the angle of the line we're graphing. Purplemath. Once you've learned the basic keywords for translating word problems from English into mathematical expressions and equations, you'll be presented with various English expressions, and be told to perform the translation. Don't view the lists of keywords as holy writ, handed down from on high. Instead, use these lists …Classify the following equations according to the type of conic each represents: A) 3 x2 + 3 y2 − 6 x + 9 y − 14 = 0. B) 6 x2 + 12 x − y + 15 = 0. C) x2 + 2 y2 + 4 x + 2 y − 27 = 0. D) x2 − y2 + 3 x − 2 y − 43 = 0. A) Both variables are squared, and both squared terms are multiplied by the same number, so this is a circle.Use completing the square to solve x2 − 4x − 8 = 0. As noted above, this quadratic does not factor, so I can't solve the equation by factoring. And they haven't given me the equation in a form that is ready to square-root. But there is a way for me to manipulate the quadratic to put it into that ready-for-square-rooting form, so I can … Pre-algebra and algebra lessons, from negative numbers through pre-calculus. Grouped by level of study. Lessons are practical in nature informal in tone, and contain many worked examples and warnings about problem areas and probable "trick" questions. The two rules for function reflection are these: To reflect the graph of a function h(x) over the x -axis (that is, to flip the graph upside-down), multiply the function by −1 to get −h(x). To reflect the graph of a function h(x) around the y -axis (that is, to mirror the two halves of the graph), multiply the argument of the function by ...For the same reason, you can take any odd root (third root, fifth root, seventh root, etc.) of a negative number. Squaring a negative number multiplies it by itself, meaning two minus signs that cancel; e.g. (−3)² …The basic metric units are meters (for length), grams (for mass or weight), and liters (for volume). And the different units convert into one another rather nicely, with one milliliter equalling one cubic centimeter (where one Cubic Centimeter is the "cc" of medical shows on television) and one gram being the mass (or weight) of one cc …Purplemath. The "addition" method of solving systems of linear equations is also called the "elimination" method. Under either name, this method is similar to the method you …In the above example, the degree on the denominator (namely, 2) was bigger than the degree on the numerator (namely, 1), and the horizontal asymptote was y = 0 (that is, it was the x-axis).This property is always true: If the degree on x in the denominator is larger than the degree on x in the numerator, then the …Purplemath. Back when you first studied square roots and how to solve radical equations, you were probably introduced to something called "the Pythagorean Theorem". This Theorem relates the lengths of the three sides of any right triangle. This Theorem existed way before Pythagorus and his followers, the … Free math problem solver answers your algebra homework questions with step-by-step explanations. Share your videos with friends, family, and the world. Kweli rapper, Michigan ann arbor news, Spokane powersports, Farmers market albuquerque, Monarch mountain lodge, Ancora coffee, University of hawaii west oahu kapolei hi, Ridley ymca, Charity navigators, North conway railroad, Crackin crab albuquerque, Romans deli, Pickle bar, Nw automotive group, Escanaba casino, Cremation society of sc, Lorie morgan, Farmers meat market.