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Chapter 6: Problem 59

In Exercises \(51-62\), factor the polynomial by grouping. $$ 6 x^{2}+3 x-2 x-1 $$

Short Answer

Expert verified
The factored form of the polynomial \(6 x^{2}+3 x-2 x-1\) is \((3x - 1)(2x + 1)\).

Step by step solution

01

Arrange the terms

First, rearrange the terms of the polynomial in such a way that pairs of terms with common factors are together. The given polynomial \(6 x^{2}+3 x-2 x-1\) can be rearranged as \(6 x^{2}-2 x + 3 x-1\). This ensures that terms with 'x' are grouped together.
02

Factor each group

Next, identify common factors and factor them out from each group. The first two terms \(6 x^{2}-2 x = 2x(3x - 1)\) and the next two terms \(3 x-1= 1(3x - 1)\). After this step, the polynomial should look like this: \(2x(3x - 1) + 1(3x - 1)\).
03

Factor the common binomial

Then identify the common binomial factor and factor it out. In this case, the term \(3x - 1\) is a common factor, and can be factored out to yield the final answer: \((3x - 1)(2x + 1)\).

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Key Concepts

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

Grouping Method
The grouping method is a handy technique for factoring polynomials, especially those with four terms. It involves rearranging and pairing the terms in the polynomial to make them easier to factor.
Think of it like grouping similar items together in a shopping basket for faster checkout. In polynomial terms, we look for pairs of terms that have a common factor.
  • The first step is to rearrange the polynomial expression. By moving terms around, you aim to create pairs or groups that have something in common. For instance, in the expression \(6x^2 + 3x - 2x - 1\), you could rearrange it to \(6x^2 - 2x + 3x - 1\).
  • In this new arrangement, the first pair \(6x^2 - 2x\) and the second pair \(3x - 1\) make it easier to spot common factors.
The goal of grouping is to simplify and to eventually find a factor common to all groups, turning a complicated polynomial into the product of simpler expressions.
Common Factors
Common factors are essential in the grouping method because they help simplify expressions. When you're factoring by grouping, you're looking for factors shared within each group.
  • For instance, consider the pair \(6x^2 - 2x\). Here, both terms share the factor \(2x\). You can factor this out to get \(2x(3x - 1)\).
  • In the pair \(3x - 1\), while it might seem that there's no obvious common factor, you can still think of it as sharing the 'factor' of 1, yielding \(1(3x - 1)\).
Once you've factored each pair as much as possible, you can then move on to finding overarching common factors across the entire expression.
Common factors make it easier to break down and understand complex polynomials, transforming them into products of simpler components.
Binomial Factor
A binomial factor in a polynomial is a factor containing exactly two terms. Identifying and factoring out a binomial is often the last step in the grouping method.
In our example expression, after grouping, \(3x - 1\) emerged as a common binomial factor. Here's how that works:
  • After factoring each group, we're left with the expression \(2x(3x - 1) + 1(3x - 1)\).
  • The term \(3x - 1\) is present in both groups as a binomial factor. So we can "factor it out" from the entire expression, much like taking a common ingredient out of two different recipes.
The result is a product of binomials: \((3x - 1)(2x + 1)\).
By factoring out the common binomial \(3x - 1\), we effectively simplify the polynomial into the product of two factors, offering a clearer picture of its structure.

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Most popular questions from this chapter

Evaluate the quantity mentally using the two samples as models. $$ \begin{array}{rlrl} 29^{2}=(30-1)^{2} & =30^{2}-2 \cdot 30 \cdot 1+1^{2} & 48 \cdot 52 & =(50-2)(50+2) \\ & =900-60+1=841 & & =50^{2}-2^{2}=2496 \end{array} $$ $$ 59 \cdot 61 $$

In Exercises 67-74, factor the polynomial completely. $$ x^{2}-81 $$

In Exercises 67-74, factor the polynomial completely. $$ 9 x^{2}+42 x+49 $$

Solve the equation. $$ 3 y^{2}-27=0 $$

Determine whether the statement is true or false. Justify your answer. $$ x^{3}-27=(x-3)^{3} $$
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