Problem 32 - Algebra for College Students

Question

For Problems \(21-34\), use the factor theorem to help answer some questions about factors. $$ \text { Is } x-2 \text { a factor of } x^{3}-8 ? $$

Step-by-Step Solution

Answer

Yes, \(x-2\) is a factor of \(x^3 - 8\).

Step 1: Understand the Factor Theorem

The Factor Theorem states that a polynomial \( f(x) \) has a factor \( x-a \) if and only if \( f(a) = 0 \). To determine if \( x-2 \) is a factor of \( x^3 - 8 \), we need to evaluate the polynomial at \( x = 2 \).

Step 2: Substitute and Evaluate

Substitute \( x = 2 \) into the polynomial \( f(x) = x^3 - 8 \). This gives us \( f(2) = 2^3 - 8 \).

Step 3: Compute the Result

Calculate \( 2^3 - 8 \). This simplifies to \( 8 - 8 = 0 \). Since \( f(2) = 0 \), it confirms that \( x-2 \) is indeed a factor.

Key Concepts

Polynomial Division

Polynomial division is a method used to divide polynomials similar to how you divide numbers. When dealing with polynomials, especially when determining whether one polynomial is a factor of another, polynomial division becomes handy. In simple terms, you perform polynomial division by dividing the terms of the polynomial in the highest degree order. This helps in reducing larger polynomials into simpler factors.

To divide polynomials, you identify the first term of the dividend by the first term of the divisor, subtract what you've found from the dividend, and repeat the process with the remainder. It resembles long division and primarily involves aligning terms according to their degree.

Polynomial division can help simplify problems in algebra by illustrating relationships between different parts of a polynomial. It's instrumental when verifying if a given expression is a zero or a factor of a polynomial using techniques like synthetic division.

Zero of a Polynomial

Zeros of a polynomial, also known as roots or solutions, are the values of the variable that make the polynomial equal to zero. These are crucial in understanding the behavior of the polynomial because they indicate where the graph of the polynomial touches or crosses the x-axis.

To find the zero of a polynomial, you set the polynomial equal to zero and solve for the variable. For example, if you have a polynomial like \( f(x) = x^3 - 8 \), you solve \( f(x) = 0 \) to find its zeros. If substituting a value of \( x \) yields zero, then that value is a zero of the polynomial.

In the problem context, using the Factor Theorem reveals that if \( f(a) = 0 \), then \( x-a \) is a factor. This theorem is practical in quickly identifying zeros and factors without full polynomial division.

Factoring Polynomials

Factoring polynomials is the process of expressing the polynomial as a product of its factors. It's a key skill in algebra that simplifies problems and solutions by breaking down complex expressions into simpler parts that are easier to work with.

To factor polynomials effectively, one often uses methods such as grouping, trial and error, or the Factor Theorem. Recognizing patterns like perfect squares or difference of squares also aids in factoring efficiently.

In the case of the Factor Theorem, to determine if \( x - a \) is a factor of a polynomial, you simply substitute \( x = a \) into the polynomial. If the result is zero, then \( x - a \) is a factor. This method helps in reducing unnecessary steps and solving polynomial expressions more expediently. Additionally, factoring helps in solving equations, simplifying expressions, and understanding polynomial functions.