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Chapter 6: Problem 105

Suppose that a polynomial contains four terms and can be factored by grouping. Explain how to obtain the factorization.

Short Answer

Expert verified
To factorize a polynomial by grouping, we first group the terms in pairs. We then find the GCF in each group, and factor it out of each group, simplifying the expressions. Using the associative property, we observe that the simplified expressions from each group now have a common factor which we can factor out. The final result is the factorization of the original polynomial.

Step by step solution

01

Grouping the Terms

We start by grouping pairs of terms. The common way to do this is in pairs that come next to each other, but this may change depending on the polynomial.
02

Factoring Common Terms

In each group that was identified in the first step, find the greatest common factor (GCF) and factor it out. Each group of terms should reduce to a simplified expression.
03

Associative Property

By examining the simplified expressions obtained in step 2, we should find that they now have a common factor. We factor this expression out using the associative property of multiplication.

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Most popular questions from this chapter

Exercises 150鈥�152 will help you prepare for the material covered in the next section. Evaluate \(2 x^{2}+7 x-4\) for \(x=\frac{1}{2}\)

Determine whether each statement 鈥渕akes sense鈥� or 鈥渄oes not make sense鈥� and explain your reasoning. The factorable trinomial \(4 x^{2}+8 x+3\) and the prime trinomial \(4 x^{2}+8 x+1\) are in the form \(a x^{2}+b x+c\) but \(b^{2}-4 a c\) is a perfect square only in the case of the factorable trinomial.

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$x^{2}-4 x y-12 y^{2}$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$4 a y^{3}-12 a y^{2}+9 a y$$

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$2 r^{3}+30 r^{2}-68 r$$
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