Intro to polynomials : Polynomial expressions, equations, & functions Adding & subtracting polynomials : Polynomial expressions, equations, & functions Adding & subtracting polynomials: two variables : Polynomial expressions, equations, & functions Multiplying monomials : Polynomial expressions, equations, & functions Multiplying monomials by polynomials : Polynomial expressions, equations, & functions Multiplying binomials : Polynomial expressions, equations, & functions Special products of binomials : Polynomial expressions, equations, & functions Multiplying binomials by polynomials : Polynomial expressions, equations, & functions Polynomials word problems : Polynomial expressions, equations, & functions Introduction to factorization : Polynomial expressions, equations, & functions Factoring monomials : Polynomial expressions, equations, & functions Factoring polynomials by taking common factors : Polynomial expressions, equations, & functions Evaluating expressions with unknown variables : Polynomial expressions, equations, & functions Factoring quadratics intro : Polynomial expressions, equations, & functions Factoring quadratics by grouping : Polynomial expressions, equations, & functions Factoring polynomials with quadratic forms : Polynomial expressions, equations, & functions Factoring quadratics: Difference of squares : Polynomial expressions, equations, & functions Factoring quadratics: Perfect squares : Polynomial expressions, equations, & functions Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License. Use the information below to generate a citation. Then you must include on every digital page view the following attribution: If you are redistributing all or part of this book in a digital format, Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses the The process will be made more clear in Example 3. In this case, the quotient is 2 x 2 – 7 x + 18 2 x 2 – 7 x + 18 and the remainder is –31. The bottom row represents the coefficients of the quotient the last entry of the bottom row is the remainder. We then multiply it by the “divisor” and add, repeating this process column by column, until there are no entries left. The process starts by bringing down the leading coefficient. Also, instead of dividing by 2, as we would in division of whole numbers, then multiplying and subtracting the middle product, we change the sign of the “divisor” to –2, multiply and add. Collapse the table by moving each of the rows up to fill any vacant spots. Synthetic division carries this simplification even a few more steps. If we don’t write the variables but, instead, line up their coefficients in columns under the division sign and also eliminate the partial products, we already have a simpler version of the entire problem. There is a lot of repetition in the table. The final form of the process looked like this: To illustrate the process, recall the example at the beginning of the section.ĭivide 2 x 3 − 3 x 2 + 4 x + 5 2 x 3 − 3 x 2 + 4 x + 5 Synthetic division is a shorthand method of dividing polynomials for the special case of dividing by a linear factor whose leading coefficient is 1. Using Synthetic Division to Divide PolynomialsĪs we’ve seen, long division of polynomials can involve many steps and be quite cumbersome. This should look familiar, since it is the same method used to check division in elementary arithmetic.ĭivide 16 x 3 − 12 x 2 + 20 x − 3 16 x 3 − 12 x 2 + 20 x − 3īy 4 x + 5. For example, let’s divide 178 by 3 using long division.Īnother way to look at the solution is as a sum of parts. We divide, multiply, subtract, include the digit in the next place value position, and repeat. We begin by dividing into the digits of the dividend that have the greatest place value. We are familiar with the long division algorithm for ordinary arithmetic. Using Long Division to Divide Polynomials To find the height of the solid, we can use polynomial division, which is the focus of this section. The length of the solid is given by 3 x 3 x For example, suppose the volume of a rectangular solid is given by the polynomial 3 x 4 − 3 x 3 − 33 x 2 + 54 x. We can also use the same method if any, or all, of the measurements contain variable expressions. We can use similar methods to find any of the missing dimensions.
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