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Ch. 3 - Polynomial and Rational Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 3 - Polynomial and Rational FunctionsProblem 24
Chapter 4, Problem 24

Use synthetic division to perform each division. x7+1 / x+1

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Identify the divisor and the dividend. Here, the divisor is \(x + 1\), and the dividend is \(x^7 + 1\).
Set up synthetic division by using the zero of the divisor. Since the divisor is \(x + 1\), the zero is \(-1\).
Write down the coefficients of the dividend polynomial \(x^7 + 1\). Since some powers of \(x\) are missing, include zeros for those terms: \$1, 0, 0, 0, 0, 0, 0, 1$.
Perform synthetic division: bring down the first coefficient, multiply it by \(-1\), add to the next coefficient, and repeat this process across all coefficients.
The numbers obtained at the end (except the last one) represent the coefficients of the quotient polynomial, and the last number is the remainder.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form x - c. It simplifies the long division process by using only the coefficients of the polynomial, making calculations faster and less error-prone.
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Polynomial Coefficients and Setup

To use synthetic division, you must list all coefficients of the dividend polynomial in descending order of degree, including zeros for any missing terms. This ensures the division process accounts for every power of x correctly.
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Interpreting the Result

The output of synthetic division includes the coefficients of the quotient polynomial and a remainder. Understanding how to write the quotient in polynomial form and interpret the remainder is essential for completing the division problem.
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Related Practice
Textbook Question

Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=4x7-x5+x3-1

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Textbook Question

Find a polynomial function ƒ(x) of least degree with real coefficients having zeros as given. √3, -√3, 2, 3

Textbook Question

Write each formula as an English phrase using the word varies or proportional. V = 1/3 πr2h, where V is the volume of a cone of radius r and height h

Textbook Question

Use one of the end behavior diagrams below, to describe the end behavior of the graph of each polynomial function.

ƒ(x)=−4x3+3x2−1 ƒ(x)=-4x^3+3x^2-1

Textbook Question

Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. ƒ(x) = (x - 2)2

Textbook Question

Solve each polynomial inequality. Give the solution set in interval notation.

(a) -x(x - 1)(x - 2) ≥ 0

(b) -x(x - 1)(x - 2) > 0

(c) -x(x - 1)(x - 2) ≤ 0

(d) -x(x - 1)(x - 2) < 0