Problem 70 - Elementary and Intermediate Algebra

Question

Factor each polynomial by factoring out the opposite of the GCF. $$ -28 a^{5}-42 a^{4}+14 a^{3} $$

Step-by-Step Solution

Answer

The polynomial factors to \(-14a^3(2a^2 + 3a - 1)\).

Step 1: Identify the Greatest Common Factor (GCF)

Start by identifying the greatest common factor of all the terms in the polynomial \(-28a^5 - 42a^4 + 14a^3\). The numerical GCF of the coefficients \(-28, -42\), and \(14\) is \(14\). Since all terms contain \(a^3\), the variable part of the GCF is \(a^3\). The overall GCF is \(14a^3\).

Step 2: Determine the Opposite of the GCF

Next, determine the opposite of the GCF. Since the GCF is \(14a^3\), the opposite is \(-14a^3\). This will be factored out from the polynomial.

Step 3: Factor out the Opposite of the GCF

Now, divide each term of the polynomial by \(-14a^3\) to factor it out:\[ -28a^5 \div (-14a^3) = 2a^2 \]\[ -42a^4 \div (-14a^3) = 3a \]\[ 14a^3 \div (-14a^3) = -1 \]This results in the expression being rewritten as:\[ -14a^3(2a^2 + 3a - 1) \]

Step 4: Verify Factoring

Check the factored expression by distributing \(-14a^3\) back into the expression \(2a^2 + 3a - 1\) and verify that it simplifies to the original polynomial:\[ -14a^3(2a^2 + 3a - 1) = -28a^5 - 42a^4 + 14a^3 \].This matches the original polynomial, confirming the factorization is correct.

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor, or GCF, refers to the largest factor shared by all terms in a polynomial. It is crucial to identify the GCF when factoring polynomials, as it helps simplify expressions effectively.

• Numerical GCF: Look at the coefficients of the terms. For the given polynomial \(-28a^5 - 42a^4 + 14a^3\), the coefficients are \(-28, -42, 14\). The GCF of these numbers is 14, as it is the largest number that evenly divides each coefficient.
• Variable GCF: Next, analyze the variable aspect. In the expression, all terms have the variable \(a\) to a certain power. The smallest power of \(a\) across the terms is \(a^3\). So, this becomes the variable component of the GCF.

Thus, the GCF for the polynomial \(-28a^5 - 42a^4 + 14a^3\) is \(14a^3\). This provides a foundation for simplifying the polynomial further by factoring out this common factor.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials. This is a fundamental skill that aids in solving equations, simplifying expressions, and understanding polynomial behavior.

• Determine the opposite of the GCF: Once you have identified the GCF, in some cases, such as when instructed to find the opposite, you will use the negative of the GCF for factoring. In this exercise, the GCF \(14a^3\) becomes \(-14a^3\).
• Divide each term by opposite GCF: For each term in the polynomial, divide by the \(-14a^3\). This simplification process separates out the common factor and gives you a cleaner expression inside the parenthesis.

After dividing \(-28a^5, -42a^4, \) and \(14a^3\) by \(-14a^3\), the polynomial is factored to \(-14a^3(2a^2 + 3a - 1)\), making it easier to handle and analyze further.

Algebra Concepts

Algebra is the broad branch of mathematics that deals with symbols and the rules for manipulating these symbols. In this context, understanding how to factor polynomials by recognizing patterns and using arithmetic operations is key.

• Verification: One significant algebraic concept is verifying your factored result. This directly involves using the distributive property to multiply back and check if you recapture the original expression. For instance, expanding \(-14a^3(2a^2 + 3a - 1)\) should return \(-28a^5 - 42a^4 + 14a^3\).
• Simplification: Factoring is a simplification strategy. It allows you to break down complex expressions into manageable parts, often simplifying problems like solving equations or integrating functions.

By honing these algebraic skills, including the ability to factor polynomials, students deepen their mathematical understanding and improve problem-solving efficiency.