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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 55
Chapter 4, Problem 55

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) = 5x4 + 2x3 -x+3; k=2/5

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Write down the coefficients of the polynomial ƒ(x) = 5x^4 + 2x^3 + 0x^2 - 1x + 3. The coefficients are 5, 2, 0, -1, and 3.
Set up synthetic division using k = \(\frac{2}{5}\). Write the coefficients in a row and place k to the left.
Bring down the first coefficient (5) as it is. Multiply this number by k (\(\frac{2}{5}\)) and write the result under the next coefficient.
Add the second coefficient (2) and the number just written. Write the sum below the line. Repeat the multiply and add process for all coefficients.
The last number you get after completing synthetic division is the remainder, which equals ƒ(k). If this remainder is 0, then k is a zero of the polynomial; otherwise, it is not.

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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 - 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 if k is a root by checking if the remainder is zero.
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Polynomial Functions and Zeros

A zero of a polynomial function is a value of x that makes the function equal to zero. If substituting k into the polynomial yields zero, then k is a root or zero of the polynomial. Identifying zeros is essential for factoring and graphing polynomial functions.
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Evaluating Polynomials at a Given Value

Evaluating a polynomial at a specific value k means substituting k into the polynomial and calculating the result. If the result is not zero, it gives the value of ƒ(k), which indicates that k is not a zero of the polynomial. Synthetic division can also provide this value as the remainder.
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Related Practice
Textbook Question

For each polynomial function, identify its graph from choices A–F. ƒ(x)=(x-2)2(x-5)2

Textbook Question

Solve each rational inequality. Give the solution set in interval notation. (2x + 3)/(x - 5) ≤ 0

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Textbook Question

Find a polynomial function ƒ(x) of degree 3 with real coefficients that satisfies the given conditions. Zeros of -2, 1, and 0; ƒ(-1)=-1

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Textbook Question

For each polynomial function, identify its graph from choices A–F.

ƒ(x)=−(x−2)2(x−5) ƒ(x)=-(x-2)^2(x-5)

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Textbook Question

Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ.

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Textbook Question

Solve each rational inequality. Give the solution set in interval notation. (3x + 7)/(x - 3) ≤ 0

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