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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 49
Chapter 4, Problem 49

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 + 4x -4; k=2

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1
Write down the coefficients of the polynomial ƒ(x) = x^3 - 3x^2 + 4x - 4. These are 1, -3, 4, and -4.
Set up the synthetic division by writing the number k = 2 to the left, and the coefficients in a row to the right: 1, -3, 4, -4.
Bring down the first coefficient (1) as it is. Then multiply this number by k (2) and write the result under the next coefficient.
Add the column values: add the second coefficient (-3) and the number you just wrote under it. Write the sum below the line.
Repeat the multiply and add process for the remaining coefficients. The last number you get after adding is the remainder, which equals ƒ(2). If this remainder is zero, then k=2 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.

Polynomial Functions

A polynomial function is an expression involving variables raised to whole-number exponents combined using addition, subtraction, and multiplication. Understanding the structure of polynomials, such as degree and coefficients, is essential for evaluating and manipulating them.
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Introduction to Polynomial Functions

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a binomial of the form (x - k). It simplifies the division process by using only the coefficients, making it faster and less error-prone than long division.
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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. Determining if k is a zero involves evaluating the polynomial at k; if the result is zero, k is a root, otherwise, the value ƒ(k) indicates the polynomial's output at that point.
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Finding Zeros & Their Multiplicity
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)

Textbook Question

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

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

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

Textbook Question

Graph each polynomial function. ƒ(x)=2x3+x2-x

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