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Chapter 1: Problem 75

Use the distributive law to factor each of the following. Check by multiplying. $$ 12 x+9 $$

Short Answer

Expert verified
The factored form is 3(4x + 3).

Step by step solution

01

Identify the greatest common factor (GCF)

First, identify the greatest common factor (GCF) of the terms in the expression. The terms are 12x and 9. The GCF of 12 and 9 is 3.
02

Factor out the GCF

Next, factor out the GCF from each term in the expression. Divide each term by the GCF (3) and place the GCF outside of the parentheses. The expression becomes: 3(4x + 3)
03

Check by distributing

To check your work, distribute the GCF. Multiply each term inside the parentheses by 3 to see if you get the original expression: 3(4x) + 3(3) = 12x + 9. The original expression is obtained, confirming that the factorization is correct.

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Key Concepts

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

distributive law
The distributive law is a fundamental principle used in algebra to simplify expressions and solve equations. It states that multiplying a single term by a sum of terms inside parentheses is the same as multiplying each individual term separately and then adding the products. This can be written as: \[a(b + c) = ab + ac\] where \(a\), \(b\), and \(c\) are any real numbers. In the example given in the exercise, the distributive law is used in the final step to check our factorization. By distributing the outside term (3) to each term inside the parentheses, we confirm our factored expression is correct.
greatest common factor
The greatest common factor, or GCF, is the largest factor that divides two or more numbers. It helps in simplifying algebraic expressions by factoring out the common terms. To find the GCF of 12 and 9 in the exercise:
  • First, list the factors of each number. Factors of 12 are 1, 2, 3, 4, 6, and 12. Factors of 9 are 1, 3, and 9.
  • Second, identify the largest common factor. In this case, the GCF is 3.
Using the GCF, we can rewrite the original terms by factoring out the GCF, which helps in simplifying our expression.
simplification
Simplification in algebra involves making an expression easier to work with by reducing it to its simplest form. This can involve factoring, combining like terms, and reducing fractions. In our example:
  • We initially had the expression 12x + 9.
  • By identifying and factoring out the GCF (3), we transformed it to 3(4x + 3).
  • This simplified form makes further operations more manageable.
Always check your simplification steps by distributing to ensure you can return to the original expression. Simplification is crucial for solving equations and making expressions easier to handle and understand.

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Most popular questions from this chapter

Simplify. Match the algebraic expression with the equivalent rewritten expression below. Check your answer by calculating the expression by hand and by using a calculator. A) \((5(3-7)+4 \wedge 3) /(-2-3)^{2}\) B) \((5(3-7)+4 \wedge 3) /\left(-2-3^{2}\right)\) C)\((5(3-7)+4) \wedge 3 /-2-3^{2}\) D) \(5(3-7)+4 \wedge 3 /(-2-3)^{2}\) $$ \frac{5(3-7)+4^{3}}{(-2-3)^{2}} $$

Write an equivalent expression without using grouping symbols. $$ -[5 x-2 y-3 z] $$

SYNTHESIS Is it true that for all real numbers \(a\) and \(b\) $$ a b=(-a)(-b) ? $$ Why or why not?

Translate to an inequality. The distance from \(t\) to 0 is at least \(20 .\)

Write an equivalent expression without using grouping symbols. $$ -(6 x-7) $$
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