Ch. 3 - Polynomial and Rational Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 51

Chapter 4, Problem 51

For each polynomial function, find all zeros and their multiplicities. $ƒ (x) = (x^{2} + x - 2)^{5} (x - 1 + \sqrt{3})^{2}$

Verified step by step guidance

Start by identifying the factors of the polynomial function: $f(x) = (x^2 + x - 2)^5 (x - 1 + \sqrt{3})^2$.

Find the zeros of the first factor $x^2 + x - 2$ by solving the quadratic equation $x^2 + x - 2 = 0$. Use factoring, completing the square, or the quadratic formula.

Once you find the roots of $x^2 + x - 2 = 0$, note that each root has a multiplicity of 5 because the entire quadratic factor is raised to the 5th power.

Next, find the zero of the second factor $x - 1 + \sqrt{3} = 0$ by isolating $x$, which gives $x = 1 - \sqrt{3}$. This zero has a multiplicity of 2 since the factor is squared.

List all zeros found along with their multiplicities: the roots from the quadratic factor each with multiplicity 5, and the root from the linear factor with multiplicity 2.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Zeros

Zeros of a polynomial are the values of x that make the polynomial equal to zero. Finding zeros involves solving the equation f(x) = 0, which can be done by factoring or using other algebraic methods. Each zero corresponds to a root of the polynomial.

Multiplicity of Zeros

Multiplicity refers to how many times a particular zero appears as a factor in the polynomial. If a factor is raised to a power n, the zero associated with that factor has multiplicity n. Multiplicity affects the graph's behavior at the zero.

Factoring and Solving Quadratic Expressions

Factoring quadratic expressions like x^2 + x - 2 helps find zeros by rewriting the polynomial as a product of linear factors. Solving these factors set to zero gives the roots. Recognizing and factoring quadratics is essential for breaking down complex polynomials.

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Related Practice

Textbook Question

Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers given. ƒ(x)=x⁴-4x³-x+3; 0.5 and 1

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) = 2x³ - 6x² -9x + 4; k=1

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Textbook Question

Work each problem. Which function has a graph that does not have a vertical asymptote?

A. ƒ(x)=1/(x²+2)

B. ƒ(x)=1/(x²-2)

C. ƒ(x)=3/x²

D. ƒ(x)=(2x+1)/(x-8)

Textbook Question

Connecting Graphs with Equations Find a quadratic function f having the graph shown. (Hint: See the Note following Example 3.)

Textbook Question

For each polynomial function, find all zeros and their multiplicities. $ƒ (x) = (2 x^{2} - 7 x + 3)^{3} (x - 2 - \sqrt{5})$

Textbook Question

Work each problem. Which function has a graph that does not have a horizontal asymptote?

A. ƒ(x)=(2x-7)/(x+3)

B. ƒ(x)=3x/(x²-9)

C. ƒ(x)=(x²-9)/(x+3)

D. ƒ(x)=(x+5)/(x+2)(x-3)

For each polynomial function, find all zeros and their multiplicities. ƒ(x)=(x2+x−2)5(x−1+3)2ƒ(x)=(x^2+x-2)^5(x-1+\(\sqrt\)3)^2

Verified video answer for a similar problem:

Key Concepts

Polynomial Zeros

Multiplicity of Zeros

Factoring and Solving Quadratic Expressions

For each polynomial function, find all zeros and their multiplicities. $ƒ (x) = (x^{2} + x - 2)^{5} (x - 1 + \sqrt{3})^{2}$