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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 39
Chapter 4, Problem 39

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

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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 substitute k into the polynomial.
Identify the polynomial and the value of k: Here, ƒ(x) = x^2 - 5x + 1 and k = 2 + i, where i is the imaginary unit.
Substitute k = 2 + i into the polynomial: Replace every x in ƒ(x) with (2 + i), so write ƒ(2 + i) = (2 + i)^2 - 5(2 + i) + 1.
Expand the squared term: Calculate (2 + i)^2 by using the formula (a + b)^2 = a^2 + 2ab + b^2, where a = 2 and b = i.
Simplify the expression by performing the multiplication and combining like terms, remembering that i^2 = -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, and multiplication, with non-negative integer exponents. Understanding the structure of polynomials helps in evaluating them at specific values, such as complex numbers.
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Introduction to Polynomial Functions

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 k by evaluating f(k) directly, without performing polynomial division.
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Higher Powers of i

Evaluating Polynomials at Complex Numbers

Evaluating a polynomial at a complex number involves substituting the complex value into the polynomial and simplifying using algebraic rules for complex numbers, including combining like terms and applying i² = -1. This is essential when k is a complex number like 2 + i.
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Dividing Complex Numbers
Related Practice
Textbook Question

For each polynomial function, one zero is given. Find all other zeros. ƒ(x)=−x4−5x2−4;−iƒ(x)=-x^4-5x^2-4; -i

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Textbook Question

Solve each polynomial inequality. Give the solution set in interval notation. 2x3 - 7x2 ≥ 3 - 8x

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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 - 2)2 - 5

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4.

ƒ(x)=x3+x2−36x−36ƒ(x)=x^3+x^2-36x-36

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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)=3/(x-5)

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=-x3+x2+2x

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