Solve each problem. Is x+1 a factor of ƒ(x)=x3+2x2+3x+2?

Ch. 3 - Polynomial and Rational Functions

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

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Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 35

Chapter 4, Problem 35

Solve each problem. Is x+1 a factor of ƒ(x)=x³+2x²+3x+2?

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Recall the Factor Theorem, which states that if $x - c$ is a factor of a polynomial $f(x)$, then $f(c) = 0$. In this problem, since the factor is $x + 1$, rewrite it as $x - (-1)$, so $c = -1$.

Evaluate the polynomial $f(x) = x^3 + 2x^2 + 3x + 2$ at $x = -1$ by substituting $-1$ into the polynomial: calculate $f(-1) = (-1)^3 + 2(-1)^2 + 3(-1) + 2$.

Simplify each term in the expression: $(-1)^3 = -1$, $2(-1)^2 = 2(1) = 2$, $3(-1) = -3$, and the constant term is $2$.

Add the simplified terms together: $-1 + 2 - 3 + 2$ to find the value of $f(-1)$.

If the result of $f(-1)$ is zero, then $x + 1$ is a factor of $f(x)$. If it is not zero, then $x + 1$ is not a factor.

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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 helps in analyzing their factors and roots.

Factor Theorem

The Factor Theorem states that (x - c) is a factor of a polynomial ƒ(x) if and only if ƒ(c) = 0. This theorem is used to test whether a given binomial is a factor by substituting the root into the polynomial.

Evaluating Polynomials

Evaluating a polynomial involves substituting a specific value for the variable and simplifying to find the result. This process is essential for applying the Factor Theorem to determine if a binomial is a factor.

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