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Chapter 5: Problem 43

Factor each polynomial by factoring out the opposite of the GCF. $$ -18 a^{2} b+12 a b^{2} $$

Short Answer

Expert verified
\(-6ab(3a - 2b)\).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

First, identify the GCF of the coefficients and the variables in the terms \(-18a^2b\) and \(12ab^2\). The GCF of the coefficients \(-18\) and \(12\) is \(6\). For the variables, both terms have at least one \(a\) and one \(b\), so the GCF in terms of variables is \(ab\). Thus, the GCF of the entire expression is \(6ab\).
02

Factor out the Opposite of the GCF

Since we are factoring out the opposite of the GCF, we take out \(-6ab\) instead of \(6ab\). Divide each term in the expression by \(-6ab\):\[-18a^2b \div (-6ab) = 3a\]\[+12ab^2 \div (-6ab) = -2b\]\.
03

Write the Factored Expression

Now that we know what each term becomes after factoring out \(-6ab\), rewrite the original polynomial as a product:\(-6ab(3a - 2b)\) is the factored form of the polynomial.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Factor
To start factoring polynomials, identifying the Greatest Common Factor (GCF) is crucial. The GCF is the largest factor that each term in the polynomial shares, both in terms of numerical coefficients and variable parts.

Here's how you can find the GCF:
  • Look at the coefficients: Find the highest number that divides both coefficients evenly. For the exercise, the coefficients are -18 and 12, and their GCF is 6.
  • Consider the variables: Check what powers of variables each term shares. Both terms in the expression have the variable 'a' and the variable 'b' at least once. Therefore, the GCF in terms of the variables is 'ab'.
Combine the numerical and variable parts to get the full GCF. In this example, the GCF is 6ab.

By identifying the GCF, we're beginning the process of simplifying the polynomial.
Variable Factorization
Variable factorization involves taking common variables out of the terms of a polynomial. Let's understand it better with the polynomial -18a虏b + 12ab虏.

In variable factorization, you follow these steps:
  • Identify the common variables in all terms. Both terms here include 'a' at least once and 'b' once.
  • Factor out these common variables to simplify the expression. In our exercise, 'ab' is factored out.
This process reduces the complexity of the original polynomial, turning it into a more manageable expression. By focusing on shared variables, you can easily rearrange and simplify mathematical expressions.
Expression Simplification
Once you have identified and factored out the GCF, the final step is expression simplification. This step transforms a complicated expression into its clean, reduced form.

Here's how it's applied to our polynomial:
  • With the GCF -6ab identified and factored out, divide each term by -6ab to see what remains: -18a虏b divided by -6ab gives 3a, and 12ab虏 divided by -6ab provides -2b.
  • Use the results to write the simplified expression: The polynomial becomes -6ab(3a - 2b).
Simplifying expressions in this way highlights the interplay between various parts of the polynomial. Thus, it becomes much easier to evaluate, solve, or further manipulate the expression in mathematical tasks.

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Most popular questions from this chapter

The following graphs of two polynomial functions \(f(x)=2 x^{3}-8 x\) and \(f(x)=2 x(x+2)(x-2)\) appear to be the same. After examining their equations, explain why we know that they are identical graphs. (Graph can't copy)

Explain why \(5 x^{2}+3 x+2=2 x(x+1)+3 x^{2}\) is not a quadratic equation.

Graph the solution set of each inequality or system of inequalities on a rectangular coordinate system. $$ x \geq 2 $$

How does one determine whether a trinomial is a perfect-square trinomial?

Solve each inequality. Graph the solution set and write it using interval notation. $$ |x|>7 $$
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