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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 37
Chapter 4, Problem 37

For each polynomial function, use the remainder theorem to find ƒ(k). ƒ(x) = x3 - 4x2 + 2x+1; k = -1

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Recall the Remainder Theorem, which states that the remainder when a polynomial ƒ(x) is divided by (x - k) is equal to ƒ(k). So, to find ƒ(k), we simply evaluate the polynomial at x = k.
Substitute k = -1 into the polynomial ƒ(x) = x^3 - 4x^2 + 2x + 1. This means replacing every x in the polynomial with -1.
Write the expression after substitution: ƒ(-1) = (-1)^3 - 4(-1)^2 + 2(-1) + 1.
Simplify each term step-by-step: calculate (-1)^3, (-1)^2, multiply by coefficients, and then combine all terms.
Sum all the simplified terms to find the value of ƒ(-1), which is the remainder when ƒ(x) is divided by (x + 1).

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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 consisting of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents. Understanding the structure of polynomials, such as terms and degrees, is essential for evaluating and manipulating these functions.
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Remainder Theorem

The Remainder Theorem states that the remainder when a polynomial f(x) is divided by (x - k) is equal to f(k). This allows us to find the value of the polynomial at x = k by evaluating f(k) directly, simplifying the process of polynomial division.
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Evaluating Polynomials at a Given Value

Evaluating a polynomial at a specific value involves substituting the value into the polynomial and performing arithmetic operations to find the result. This skill is crucial for applying the Remainder Theorem and understanding the behavior of polynomial functions at particular points.
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Introduction to Polynomials
Related Practice
Textbook Question

For each polynomial function, one zero is given. Find all other zeros. ƒ(x)=4x3+6x2−2x−1; 12ƒ(x)=4x^3+6x^2-2x-1;\(\text{ }\]\frac\)12

Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=(4x+3)(x+2)2

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=x3+5x2-x-5

Textbook Question

For each polynomial function, use the remainder theorem to find ƒ(k). ƒ(x) = - x3 + 8x2 + 63; k=4

Textbook Question

Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. ƒ(x) = (x + 3)2

Textbook Question

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=3/(x-5)

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