Problem 71 - Elementary and Intermediate Algebra

Question

Factor each polynomial by factoring out the opposite of the GCF. $$ -4 a^{3} b^{2}+14 a^{2} b^{2}-10 a b^{2} $$

Step-by-Step Solution

Answer

Factor: \(-2b^{2}(2a^{3}-7a^{2}+5a)\).

Step 1: Identify the GCF

First, identify the greatest common factor (GCF) of all the terms in the polynomial. We have the terms \(-4a^{3}b^{2}, 14a^{2}b^{2},\) and \(-10ab^{2}\). In these terms, the common factor is \(b^{2}\) and the numerical coefficients \(-4, 14,\) and \(-10\) have a GCF of \(2\). Therefore, the GCF of the entire polynomial is \(2b^{2}\).

Step 2: Factor out the Positive GCF

Write the polynomial with the identified GCF factored out. The positive GCF factored polynomial is:\[2b^{2}(-2a^{3}+7a^{2}-5a)\]. This step shows the expression when the GCF is taken out but with its positive value.

Step 3: Factor out the Opposite GCF

Instead of factoring out the positive GCF, factor out its opposite, which is \(-2b^{2}\). Factoring out means dividing each term by \(-2b^{2}\):\[ -2b^{2}(2a^{3}-7a^{2}+5a) \]. This allows us to see the polynomial with its terms rearranged with opposite signs, effectively flipping the signs of each term in the expression within the parenthesis.

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is a key component when it comes to simplifying and factoring polynomials. It represents the largest factor that is shared among all the terms of a polynomial expression. For students looking into this, there are a few steps to identify the GCF efficiently:

• First, look at each term in the polynomial individually and list out the factors for both numerical coefficients and variables.
• For the numbers, find the biggest number that can divide each coefficient without leaving a remainder. In the example, \(\ -4, 14,\ \text{and } -10\) have a GCF of 2.
• For the variables, take the variable listed with the smallest power present across all terms. In our example, \(b^2\) is the common factor, as it appears with at least that power in every term.

Together, these give us the overall GCF, which in this case combines to be \(2b^2\). Proper identification of the GCF is the first step toward simplifying and rewriting polynomial expressions.

Factoring Out

Factoring out is the process of extracting the greatest common factor from the terms of a polynomial. This means you'll be essentially reversing the distributive property, dividing each term by the GCF, and writing them so that the polynomial appears in a simpler form.

When the instruction involves factoring out the opposite of the GCF, it implies taking the negative version of the GCF. This step involves:

• Dividing every term by the negative GCF.
• Changing the signs of the resulting terms inside the parentheses.
• Rewriting the expression with these new terms grouped together.

In our exercise, we move from \(2b^2(-2a^3+7a^2-5a)\) to \(-2b^2(2a^3-7a^2+5a)\). This flipping of signs is critical as it can often make the polynomial easier to work with later, especially if further factoring is required or specific simplifications are necessary.

Polynomial Expressions

Polynomials are algebraic expressions consisting of variables and coefficients, exhibiting terms separated by addition or subtraction. Polynomial expressions can be simplified in a variety of ways, with factoring being one of the most common methods.

These expressions can range in complexity. A polynomial could be as simple as \(x+2\) or as complex as the one in our exercise: \(-4a^3b^2 + 14a^2b^2 - 10ab^2\). When working with polynomials, always remember:

• Each term in a polynomial can have different variables and powers, but when factoring, identifying common elements across terms is vital.
• Maintaining the order of operations – factor the GCF, simplify, and rearrange – ensures your polynomial remains equivalent throughout.

Understanding polynomials at their core enables you to assess situations where you can simplify or factor the expression effectively. This fundamental understanding of polynomials is essential not just for factoring, but also for more advanced mathematical operations like solving equations and graphing curves.