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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 47
Chapter 4, Problem 47

Use synthetic division to determine whether the given number k is a zero of the polynomial function. If it is not, give the value of ƒ(k). ƒ(x) = x2 +2x -8; k=2

Verified step by step guidance
1
Write down the coefficients of the polynomial ƒ(x) = x^2 + 2x - 8. These are 1 (for x^2), 2 (for x), and -8 (constant term).
Set up the synthetic division by writing the number k = 2 to the left, and the coefficients 1, 2, and -8 in a row to the right.
Bring down the first coefficient (1) directly below the line. Then multiply this number by k (2) and write the result under the next coefficient.
Add the numbers in the second column (2 + the product from the previous step), write the sum below the line, then repeat the multiply and add process for the last coefficient.
The final number you get after the last addition is the remainder, which equals ƒ(k). If this remainder is 0, then k is a zero of the polynomial; if not, this remainder is the value of ƒ(k).

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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 binomial of the form (x - k). It simplifies the long division process by using only the coefficients of the polynomial, making it faster and less error-prone. This method helps determine the remainder when the polynomial is evaluated at k.
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Zeros of a Polynomial

A zero of a polynomial is a value of x that makes the polynomial equal to zero. If k is a zero, then (x - k) is a factor of the polynomial, and the remainder after division by (x - k) is zero. Identifying zeros is essential for factoring and solving polynomial equations.
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Evaluating Polynomial Functions

Evaluating a polynomial function at a specific value k means substituting k into the polynomial and calculating the result. If the result is zero, k is a root; otherwise, the value gives the output of the function at k. Synthetic division provides a quick way to find this value.
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Related Practice
Textbook Question

For each polynomial function, find all zeros and their multiplicities. ƒ(x)=(x+1)2(x−1)3(x2−10)ƒ(x)=(x+1)^2(x-1)^3(x^2-10)

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Textbook Question

Solve each polynomial inequality. Give the solution set in interval notation. x4 + 6x2 + 1 ≥ 4x3 + 4x

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Textbook Question

Graph each polynomial function. ƒ(x)=(x-2)2(x+3)

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Textbook Question

Several graphs of the quadratic function ƒ(x) = ax2 + bx + c are shown below. For the given restrictions on a, b, and c, select the corresponding graph from choices A–F. (Hint: Use the discriminant.) a < 0; b2 - 4ac < 0

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Textbook Question

Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers given. ƒ(x)=3x2-x-4; 1 and 2

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Textbook Question

Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers given. ƒ(x)=-2x3+5x2+5x-7; 0 and 1

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