Problem 22 - Algebra for College Students

Question

Factor each of the following polynomials completely. Indicate any that are not factorable using integers. Don't forget to look first for a common monomial factor. $$8 x^{2}-72$$

Step-by-Step Solution

Answer

The polynomial completely factors to \(8(x - 3)(x + 3)\).

Step 1: Look for a Common Monomial Factor

First, we need to identify a common factor in the terms of the polynomial. The polynomial given is \(8x^2 - 72\). Both terms share a factor of 8. So, we factor out 8 from the polynomial, which results in \(8(x^2 - 9)\).

Step 2: Check for Further Factorization

After factoring out the common monomial factor, we are left with \(x^2 - 9\) inside the parentheses. Recognize this as a difference of squares, since it can be expressed as \(x^2 - 3^2\).

Step 3: Factor the Difference of Squares

The expression \(x^2 - 9\) is a difference of squares and can be factored using the formula \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a = x\) and \(b = 3\), so \(x^2 - 9 = (x - 3)(x + 3)\).

Step 4: Write the Completely Factorized Form

Substituting the factored form of \(x^2 - 9\) back into our expression, the polynomial \(8x^2 - 72\) is completely factorized as \(8(x - 3)(x + 3)\).

Key Concepts

Common Monomial Factor

When factoring polynomials, the first step is always to look for a Common Monomial Factor. This means identifying a single number or variable (or a combination of both) that can divide each term of the polynomial without leaving a remainder. By factoring out the Common Monomial Factor, you simplify the polynomial and make it easier to handle.

• For the polynomial \(8x^2 - 72\), the common factor is 8. Both terms, \(8x^2\) and \(-72\), are divisible by 8.
• Once you identify the factor, you divide each term by this factor and write it outside of the parentheses: \(8(x^2 - 9)\).

When done correctly, factoring out the common factor should reduce complexity, allowing for further simplification if needed. This step is fundamental as it sets the stage for further factorization techniques.

Difference of Squares

After factoring out the Common Monomial Factor, the next step often involves recognizing special patterns, such as the Difference of Squares. The expression \(x^2 - 9\) is a classic example of this pattern since it follows the form \(a^2 - b^2\).

• The Difference of Squares pattern is characterized by two squared terms separated by a minus sign.
• This pattern can be rewritten using the formula \(a^2 - b^2 = (a - b)(a + b)\).
• In \(x^2 - 9\), let \(a = x\) and \(b = 3\), since \(9\) is \(3^2\).
• Apply the formula to get \((x - 3)(x + 3)\).

This technique is particularly useful because it allows you to break down quadratic polynomials into simpler, linear factors. Recognizing the Difference of Squares can drastically simplify the factorization process.

Polynomial Factorization Steps

Factoring polynomials is a process that requires careful attention to detail, combined with efficient techniques. Following systematic Polynomial Factorization Steps assures precision and simplicity in finding the fully factored form.

• **Begin** by identifying and factoring out any Common Monomial Factor from the polynomial. This initial step simplifies the expression.
• **Next**, scrutinize the remaining polynomial for recognizable patterns, such as the Difference of Squares. Utilize these patterns to further factor the polynomial accurately.
• **Finally**, combine all factored elements to express the original polynomial in its completely factorized form.

For example, with \(8x^2 - 72\), we initially factor out an 8, leaving \(x^2 - 9\). Recognizing this as a Difference of Squares leads us to factor it into \((x - 3)(x + 3)\). Thus, the fully factorized polynomial is \(8(x - 3)(x + 3)\). Sticking to this structure enhances clarity and reduces the risk of errors in polynomial factorization.