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Chapter 6: Problem 53

In Exercises \(51-62\), factor the polynomial by grouping. $$ x^{2}+3 x+4 x+12 $$

Short Answer

Expert verified
The factored form of the polynomial \(x^{2} + 3x + 4x + 12\) by grouping is \((x + 3)(x + 4)\).

Step by step solution

01

Regrouping

Rearrange and group the polynomial expression in such a way that there are common factors in each group. The given polynomial can be regrouped as \(x^2 + 3x\) and \(4x + 12\).
02

Find common factors

Find common factors from each group: for \(x^2 + 3x\) common factor is \(x\), so it becomes \(x(x + 3)\). For \(4x + 12\), common factor is \(4\) and it becomes \(4(x + 3)\).
03

Common factor across groups

Same factor \((x + 3)\) across both groups, hence the factored result of the given polynomial becomes \((x + 3)(x + 4)\).

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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 used in factoring polynomials, especially when dealing with expressions that have four terms. The essence of this method is to rearrange and group terms in ways that make it easier to identify common factors.
To apply the grouping method, think about how you might strategically pair terms. In the expression \( x^2 + 3x + 4x + 12 \), we can group the first two terms \( (x^2 + 3x) \) and the last two terms \( (4x + 12) \).
  • The goal here is to create sub-expressions within the polynomial that have observable and extractable common factors.
  • Once grouped, you move on to factoring out the common factors from each group.

This approach is particularly useful when direct factoring techniques aren't obvious, as it simplifies the polynomial into smaller, easier-to-manage parts.
Common factors
A common factor is a term that is shared by each component of a group. Identifying common factors is a critical step in the polynomial factoring process as it helps in breaking down the expression into simpler factors.
In our example of \( x^2 + 3x + 4x + 12 \), each group \( (x^2 + 3x) \) and \( (4x + 12) \) has a common factor.
  • For \( x^2 + 3x \), the common factor is \( x \), turning it into \( x(x + 3) \).
  • For \( 4x + 12 \), the common factor is \( 4 \), turning it into \( 4(x + 3) \).

Identifying common factors helps streamline the simplification process and sets the stage for finding a common factor between the groups themselves. This allows for a polynomial to be reduced into a product of simpler binomials or factors.
Factoring polynomials
Factoring polynomials involves expressing a polynomial as the product of its factors. It is a fundamental skill in algebra that simplifies expressions and solves for roots or zeros.
In this exercise, we have successfully applied the grouping method and identified common factors, which allow us to rewrite the polynomial in its factored form.
  • After factoring \( x(x + 3) \) and \( 4(x + 3) \), notice that \((x + 3)\) is common across both groups.
  • This enables us to group the factors as \((x + 3)(x + 4)\) by factoring out \((x + 3)\).

The final product, \((x + 3)(x + 4)\), confirms that the original polynomial can be entirely decomposed into the multiplication of simple binomial terms. Factoring provides a clearer insight into the behavior of polynomial functions and is a powerful tool in algebraic manipulation and solving equations.

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Most popular questions from this chapter

In Exercises \(45-52\), solve the equation. $$ 5 x^{2}-24=37 x $$

In Exercises 79-82, evaluate the quantity mentally using the two samples as models. $$ \begin{array}{rlrl} 29^{2}=(30-1)^{2} & =30^{2}-2 \cdot 30 \cdot 1+1^{2} & 48 \cdot 52 & =(50-2)(50+2) \\ & =900-60+1=841 & & =50^{2}-2^{2}=2496 \end{array} $$ $$ 21^{2} $$

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