Skip to main content
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 109
Chapter 4, Problem 109

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x4+2x2+1

Verified step by step guidance
1
Recognize that the polynomial ƒ(x) = x^4 + 2x^2 + 1 is a quartic polynomial, but it can be treated as a quadratic in terms of x^2. To do this, let \( y = x^2 \). Then rewrite the polynomial as \( y^2 + 2y + 1 \).
Notice that the quadratic in \( y \) is \( y^2 + 2y + 1 \), which can be factored or recognized as a perfect square trinomial. Factor it as \( (y + 1)^2 \).
Set the factored form equal to zero to find the zeros in terms of \( y \): \( (y + 1)^2 = 0 \) which implies \( y + 1 = 0 \). Solve for \( y \) to get \( y = -1 \).
Recall that \( y = x^2 \), so substitute back to get \( x^2 = -1 \). To find \( x \), take the square root of both sides, remembering to include both positive and negative roots: \( x = \pm \sqrt{-1} \).
Since \( \sqrt{-1} = i \) (the imaginary unit), the complex zeros are \( x = i \) and \( x = -i \). Because the factor was squared, each zero has multiplicity 2, so list each zero twice.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

Key Concepts

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

Complex Zeros of Polynomial Functions

Complex zeros are values of x, possibly including imaginary numbers, that make the polynomial equal to zero. Finding all complex zeros involves solving the polynomial equation, which may require factoring or using formulas, and includes real and non-real solutions.
Recommended video:
05:33
Complex Conjugates

Factoring Polynomials

Factoring breaks down a polynomial into simpler polynomials whose product equals the original. For quartic polynomials like x⁴ + 2x² + 1, recognizing patterns such as quadratic forms or perfect squares helps simplify the problem and find zeros more easily.
Recommended video:
Guided course
07:30
Introduction to Factoring Polynomials

Quadratic Substitution Method

This method involves substituting a variable (e.g., y = x²) to transform a higher-degree polynomial into a quadratic form. Solving the quadratic in y and then back-substituting helps find the original variable's zeros, including complex ones.
Recommended video:
04:03
Choosing a Method to Solve Quadratics
Related Practice
Textbook Question

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x6-9x4-16x2+144

Textbook Question

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x4+4x3+6x2+4x+1

Textbook Question

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x4-6x3+7x2

1
views
Textbook Question

Find a rational function ƒ having a graph with the given features.

x-intercepts: (1, 0) and (3, 0)

y-intercept: none

vertical asymptotes: x=0 and x=2

horizontal asymptote: y=1

Textbook Question

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x4-8x3+29x2-66x+72

2
views
Textbook Question

Find a rational function ƒ having a graph with the given features.

x-intercepts: (-1, 0) and (3, 0)

y-intercept: (0, -3)

vertical asymptote: x=1

horizontal asymptote: y=1

1
views