Problem 27 - Algebra for College Students

Question

For Problems \(21-34\), use the factor theorem to help answer some questions about factors. $$ \text { Is } x+2 \text { a factor of } x^{3}+7 x^{2}+x-18 ? $$

Step-by-Step Solution

Answer

Yes, \( x + 2 \) is a factor of the polynomial.

Step 1: Understand the Factor Theorem

The Factor Theorem states that for a polynomial \( f(x) \), \( x - r \) is a factor if and only if \( f(r) = 0 \). In this exercise, we need to determine if \( x+2 \) is a factor of \( x^{3}+7x^{2}+x-18 \), which means we will test \( f(-2) \).

Step 2: Substitute the Value into the Polynomial

We need to substitute \( x = -2 \) into the polynomial \( f(x) = x^3 + 7x^2 + x - 18 \). This means we will calculate \( f(-2) = (-2)^3 + 7(-2)^2 + (-2) - 18 \).

Step 3: Calculate the Value of the Polynomial at \( x = -2 \)

Calculate each term: \((-2)^3 = -8\), \(7(-2)^2 = 28\), \(-2 = -2\), and \(-18 = -18\). Then add these values: \[-8 + 28 - 2 - 18 = 0\].

Step 4: Determine if \( x+2 \) is a Factor

Since \( f(-2) = 0 \), according to the Factor Theorem, \( x + 2 \) is a factor of the polynomial \( x^3 + 7x^2 + x - 18 \).

Key Concepts

Polynomial Division

Polynomial division is a method used to divide polynomials, similar to how we divide numbers. It helps to break down complex polynomials into simpler factors or parts. This process can be useful when simplifying expressions or solving polynomial equations. In polynomial division, the polynomial you're dividing is called the dividend, and the polynomial by which you're dividing is called the divisor. The goal is to find the quotient and possibly a remainder, much like ordinary long division with numbers. When performing polynomial division:

• Arrange the polynomial terms in decreasing order of their exponents.
• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by this result, and subtract it from the original dividend to form a new dividend.
• Repeat the process until the degree of the remainder is less than the degree of the divisor.

Understanding polynomial division is crucial for more advanced topics like synthetic division and applying the Factor Theorem.

Factor of a Polynomial

A factor of a polynomial is a polynomial that divides the original polynomial without leaving a remainder. Finding factors is crucial because it helps simplify polynomials and solve equations where the polynomial equals zero.Using the Factor Theorem is a common way to determine factors. According to the theorem, if substituting a certain number into the polynomial results in zero, then the expression corresponding to that number is a factor:

• If \( f(r) = 0 \), then \( x - r \) is a factor of the polynomial \( f(x) \).

This is useful in, for example, determining if \( x + 2 \) is a factor. By calculating \( f(-2) \), if the result is zero, then \( x + 2 \) is confirmed as a factor.Finding all factors is often necessary for solving polynomial equations and simplifying algebraic expressions.

Evaluating Polynomials

Evaluating polynomials involves calculating the value of the polynomial for a given value of the variable. This is a straightforward process, which involves simply substituting the given value for the variable and performing the necessary mathematical operations.To evaluate a polynomial like \( f(x) = x^3 + 7x^2 + x - 18 \), you can:

• Substitute the value you're given for \( x \). For example, if you need to evaluate \( f(-2) \), substitute \( -2 \) for each \( x \) in the expression.
• Apply arithmetic operations - calculate powers, multiply coefficients, and then add or subtract the terms together.

This evaluation often determines specific properties of the polynomial, like finding roots or verifying factors as per the Factor Theorem, which states that a root \( r \) satisfies \( f(r) = 0 \).