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Chapter 3: Problem 36

For Problems \(25-50\), factor completely. $$ 24 a^{3} b^{2}+36 a^{2} b $$

Short Answer

Expert verified
The expression factors as \( 12a^2b(2ab + 3) \).

Step by step solution

01

Identify Common Factors

The first step in factoring is to identify any common factors in all the terms of the expression. Both terms of the expression, \(24a^3b^2\) and \(36a^2b\), share common factors of \(12a^2b\).
02

Factor Out the Greatest Common Factor (GCF)

Factor the greatest common factor \(12a^2b\) out of each term.\[ 24a^3b^2 + 36a^2b = 12a^2b(2ab) + 12a^2b(3) \]
03

Simplify the Expression Inside the Parentheses

Simplify the expression by combining the terms in parentheses:\[ 12a^2b(2ab + 3) \]
04

Check the Factoring

Verify the factoring by redistributing the terms:\[ 12a^2b(2ab) + 12a^2b(3) = 24a^3b^2 + 36a^2b \] If the original expression is obtained, the factoring is correct.

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Key Concepts

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

Greatest Common Factor (GCF)
When simplifying algebraic expressions, finding the greatest common factor (GCF) is a crucial step. The GCF is the largest factor shared by all the terms within a polynomial. It is similar to the GCF in basic arithmetic but applied to algebraic expressions.
The GCF can include both numbers and variables. To find the GCF:
  • Identify the greatest number that divides all coefficients.
  • Determine the lowest power of each shared variable among terms.
In our example, the expression is \(24a^3b^2 + 36a^2b\). Here, both terms have a numerical factor of \(12\), and common variable factors of \(a^2\) and \(b\). Thus, the GCF is \(12a^2b\).
Identifying the GCF helps simplify the polynomial by allowing us to factor it out, making other steps in solving equations easier.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators (such as + or -). They express relationships between quantities and are foundational to algebra. Without equality signs, they don't stand as equations but are a part of many algebraic equations.
An expression like \(24a^3b^2 + 36a^2b\) consists of terms. Each term is made up of constants (like 24 and 36) and variables (like \(a\) and \(b\)). These terms are combined using operators, forming a more complex expression.
Understanding the structure of algebraic expressions is essential. It helps in both simplifying and factoring, particularly when making calculations more manageable.
Polynomial Simplification
Simplifying polynomials involves reducing them to their simplest form without changing their value. The process often includes factoring and combining like terms.
In the exercise, we simplify \(24a^3b^2 + 36a^2b\) by factoring out the GCF identified earlier, \(12a^2b\). This results in:
  • Factoring out: \(12a^2b(2ab + 3)\).
Inside the parenthesis, \(2ab + 3\) becomes our simplified expression.
Always verify by distributing back to ensure the expression remains equivalent to the original. Simplification not only makes expressions easier to work with but also paves the way for solving more complex algebraic equations efficiently.

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Most popular questions from this chapter

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ w^{2}-4 w=5 $$

Consider the following approach to factoring the problem $$\begin{aligned} &(x-2)^{2}+3(x-2)-10 \\ &(x-2)^{2}+3(x-2)-10 \\ &=y^{2}+3 y-10 \\ &=(y+5)(y-2) \\ &=(x-2+5)(x-2-2) \\ &=(x+3)(x-4) \end{aligned}$$ Use this approach to factor Problems \(110-115\). $$ 6(x-4)^{2}+7(x-4)-3 $$

For Problems \(1-54\), solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. (Objective 1) $$ x^{4}-9 x^{2}=0 $$

Find two integers whose product is 105 such that one of the integers is one more than twice the other integer.

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) $$ n^{3}-49 n $$
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