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Chapter 3: Problem 47

For Problems \(25-50\), factor completely. $$ 3 x(2 a+b)-2 y(2 a+b) $$

Short Answer

Expert verified
The expression factors to \((2a+b)(3x-2y)\).

Step by step solution

01

Identify the Common Factor

Observe that both terms in the expression \(3x(2a+b) - 2y(2a+b)\) contain a common factor, which is \((2a+b)\). We will factor this out to simplify the expression.
02

Factor Out the Common Term

Factor \((2a+b)\) out of both terms. The expression becomes \((2a+b)(3x - 2y)\).
03

Verify the Factored Expression

Verify by distributing \((2a+b)\) back into \((3x - 2y)\) to ensure it equals the original expression. Each term should match: \((2a+b)(3x) = 3x(2a+b)\) and \((2a+b)(-2y) = -2y(2a+b)\). This confirms the factoring is correct.

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Key Concepts

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

Understanding the Common Factor
In algebra, a common factor is a term that appears in each part of an expression. Identifying common factors can simplify algebraic expressions significantly.
  • Look for repeated elements in each term.
  • In the expression \(3x(2a+b) - 2y(2a+b)\), the term \((2a+b)\) is present in both parts.
  • By recognizing this common factor, we can group and factor them out.
Factoring out common elements helps reduce complex expressions into simpler forms. Hence, simplifying problems in algebra.
Algebraic Expressions: Breaking Down the Basics
Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They form the basis of algebra and help us express relationships and changes symbolically.
  • Terms like \(3x\) and \(-2y\) in the expression are parts of algebraic expressions.
  • These terms can include coefficients, variables, and sometimes constants.
  • Understanding each component is key to manipulating expressions effectively.
By mastering the breakdown of these expressions, students can easily work on more complex algebra problems.
Simplification Made Easy
Simplification refers to reducing the expression to its simplest form. It involves making expressions as concise as possible without changing their values.
  • Factoring is a powerful method for simplification.
  • For example, simplifying \(3x(2a+b) - 2y(2a+b)\) involves factoring out the common part \((2a+b)\).
  • After factoring, we simplify the expression down to \((2a+b)(3x - 2y)\).
Simplification makes expressions easier to work with and interpret, essential in both solving and understanding algebraic equations.
Grasping the Distributive Property
The distributive property is an important algebraic rule that allows for the multiplication of a number by a group of terms inside parentheses.- It's expressed as \(a(b + c) = ab + ac\).- In our original expression, \(3x(2a+b) - 2y(2a+b)\), you apply the distributive property by factoring \((2a+b)\) out of both terms.- This illustrates how the distributive property helps simplify expressions and solve problems efficiently.Understanding the distributive property is crucial for validating the steps during simplification and ensuring the accuracy of factored forms.

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Most popular questions from this chapter

Problems \(63-100\) should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. (Objective 4) $$ 18 x^{2}-12 x+2 $$

For Problems 104-109, factor each trinomial and assume that all variables that appear as exponents represent positive integers. $$ 20 x^{2 n}+21 x^{n}-5 $$

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ 8 n^{2}-47 n-6=0 $$

Consider the following approach to factoring $$ 12 x^{2}+54 x+60: $$ $$ \begin{aligned} 12 x^{2}+54 x+60 &=(3 x+6)(4 x+10) \\ &=3(x+2)(2)(2 x+5) \\ &=6(x+2)(2 x+5) \end{aligned} $$ Is this a correct factoring process? Do you have any suggestion for the person using this approach?

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ 3 t(t-4)=0 $$
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