Problem 68 - Elementary and Intermediate Algebra

Question

Factor each polynomial by factoring out the opposite of the GCF. $$ -25 x^{4}+30 x^{2} $$

Step-by-Step Solution

Answer

The polynomial factors to -5x²(5x² - 6).

Step 1: Identify the Greatest Common Factor (GCF)

First, identify the GCF of the terms in the polynomial (-25x^4 + 30x^2). The coefficients are -25 and 30. The GCF of these coefficients is 5. Furthermore, in terms of the variable x, the smallest power of x present in both terms is x². Thus, the GCF of the entire expression is 5x².

Step 2: Factor out the GCF

Next, we need to factor out the opposite of the GCF from the polynomial. The opposite of the GCF, 5x², is -5x². Factor -5x² from each term in the polynomial to get (-5x²(-5x²) + 6) = -5x²(5x² - 6).

Step 3: Verify the Factored Expression

Verify by expanding (-5x²(5x² - 6)) to check if it matches the original polynomial. Expanding gives: -5x² * 5x² = -25x⁴ and -5x² * -6 = 30x². The results match the original polynomial (-25x⁴ + 30x²). Thus, the factorization is correct.

Key Concepts

Greatest Common Factor

When dealing with polynomials, factoring out the Greatest Common Factor (GCF) is often the first step. The GCF is the largest product of integers and variables that divides each term in the polynomial without leaving a remainder. In the exercise, we began by identifying the coefficients of the terms: -25 and 30.

• For the coefficients: the GCF is the largest positive integer that divides both numbers, which is 5.
• For the variables: the GCF is the smallest power of the variable common to each term, which is x² in this case because x² is present in both terms.

By identifying 5x² as the GCF, we establish a foundation for simplifying the polynomial expression by taking this common factor out from each term.
Factoring out the GCF decreases the complexity of the polynomial, making it easier to handle, and is a necessary step before moving towards further polynomial factorization.

Polynomial Factorization

Once the GCF is identified, the next step is to factor it out of the polynomial. However, in this particular case, we focus on factoring out the opposite of the GCF, which adds an additional challenge. Factoring the opposite of the GCF means you multiply the GCF by -1.
So here, instead of factoring out 5x², we factor out -5x². This changes the signs of the terms inside the parentheses. By dividing each term of the polynomial by -5x², we reform the expression into a product of -5x² and another polynomial:

• For the term -25x⁴: \(-25x^4 \div -5x^2 = 5x^2\)
• For the term +30x²: \(30x^2 \div -5x^2 = -6\)

Thus, the factorization results in \[-5x^{2}(5x^{2} - 6)\].This process simplifies the polynomial into a product of factors, making it easier to work with and analyze.

Mathematical Verification

Verifying your factorized expression is crucial to ensure its accuracy. In mathematics, a solution is not complete without verification. Once we factor the polynomial, it’s important to expand the factors back to confirm they form the original expression.
Here's how the verification process works for our factorized polynomial:

• Multiply each term in the factor \( (5x^2 - 6) \) by -5x², the factor that was removed initially:
• -5x² * 5x² = -25x⁴
• -5x² * -6 = 30x²

When these terms are added together, we indeed regain the original polynomial \(-25x^4 + 30x^2\).
This confirmation step helps guarantee that the factorization process was executed correctly. Confidence in your answer allows you to proceed with further mathematical operations or applications, knowing your initial step is accurate.