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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 61
Chapter 4, Problem 61

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) = 4x4 + x2 + 17x + 3; k= -3/2

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Write down the coefficients of the polynomial ƒ(x) = 4x^4 + 0x^3 + 1x^2 + 17x + 3. Note that the coefficient of x^3 is 0, so the coefficients are: 4, 0, 1, 17, 3.
Set up synthetic division using k = -\(\frac{3}{2}\). Write the coefficients in a row and place k to the left.
Bring down the first coefficient (4) as it is. Then multiply this number by k (4 \(\times\) -\(\frac{3}{2}\)) and write the result under the next coefficient.
Add the second coefficient (0) and the number just written, then repeat the multiply and add process for all coefficients.
After completing the synthetic division, check the final number (the remainder). If it is zero, 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 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 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 substituting k into the polynomial results in zero, then k is a root or zero of the polynomial. Identifying zeros is essential for factoring and solving polynomial equations.
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Evaluating Polynomials

Evaluating a polynomial at a specific value k means substituting k into the polynomial and calculating the result. If the result is zero, k is a zero of the polynomial; otherwise, the result is the value of the polynomial at k, denoted as ƒ(k). Synthetic division provides a quick way to find this value.
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Related Practice
Textbook Question

Find a polynomial function ƒ(x) of least degree having only real coefficients and zeros as given. Assume multiplicity 1 unless otherwise stated. 5+i and 5-i

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Textbook Question

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6.

ƒ(x)=3x4+2x3−4x2+x−1ƒ(x)=3x^4+2x^3-4x^2+x-1; no real zero less than -2

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Textbook Question

Show that the real zeros of each polynomial function satisfy the given conditions. ƒ(x)=3x4+2x3-4x2+x-1; no real zero greater than 1

Textbook Question

Graph each rational function. ƒ(x)=(x+1)/(x-4)

Textbook Question

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) = x3 + 3x2 -x + 1; k = 1+i

Textbook Question

Solve each rational inequality. Give the solution set in interval notation. 1 /(x+ 2) > 1 /(x -3)