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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 45
Chapter 4, Problem 45

For each polynomial function, use the remainder theorem to find ƒ(k). ƒ(x) = 6x4 + x3 - 8x2 + 5x+6; k=1/2

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1
Recall the Remainder Theorem: For a polynomial ƒ(x), the remainder when divided by (x - k) is equal to ƒ(k). So, to find ƒ(k), we simply evaluate the polynomial at x = k.
Write down the polynomial function and substitute x with k = \(\frac{1}{2}\): \[ƒ\left(\frac{1}{2}\right) = 6\left(\frac{1}{2}\right)^4 + \left(\frac{1}{2}\right)^3 - 8\left(\frac{1}{2}\right)^2 + 5\left(\frac{1}{2}\right) + 6\]
Calculate each term separately: - Compute \(6\left(\frac{1}{2}\right)^4\) - Compute \(\left(\frac{1}{2}\right)^3\) - Compute \(-8\left(\frac{1}{2}\right)^2\) - Compute \(5\left(\frac{1}{2}\right)\) - The constant term is 6
Add all the computed values together to find the value of ƒ\(\left\)(\(\frac{1}{2}\)\(\right\)). This sum represents the remainder when ƒ(x) is divided by \(x - \frac{1}{2}\).
Thus, the value of ƒ(k) is the remainder according to the Remainder Theorem, completing the problem.

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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, and multiplication, with non-negative integer exponents. Understanding the structure of polynomials helps in evaluating them at specific values and applying theorems related to their behavior.
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Introduction to Polynomial Functions

Remainder Theorem

The Remainder Theorem states that when a polynomial ƒ(x) is divided by (x - k), the remainder is equal to ƒ(k). This allows us to find the value of the polynomial at k by simply evaluating ƒ(k), without performing full polynomial division.
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Evaluating Polynomials at a Given Value

Evaluating a polynomial at a specific value involves substituting the value for the variable and simplifying the expression. This process is essential for applying the Remainder Theorem and finding the remainder or function value efficiently.
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Guided course
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Introduction to Polynomials
Related Practice
Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=3x4-7x3-6x2+12x+8

Textbook Question

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=(4x2+25)/(x2+9)

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=x4+3x3-3x2-11x-6

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Textbook Question

If the given term is the dominating term of a polynomial function, what can we conclude about each of the following features of the graph of the function? (a) domain (b) range (c) end behavior (d) number of zeros (e) number of turning points 10x7

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Textbook Question

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=(x2+1)/(x2+9)

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Textbook Question

If the given term is the dominating term of a polynomial function, what can we conclude about each of the following features of the graph of the function? (a) domain (b) range (c) end behavior (d) number of zeros (e) number of turning points -9x6

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