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Chapter 5: Problem 46

Factor expression completely. If an expression is prime, so indicate. \(60 q^{2} r^{2} s^{4}+78 q r^{2} s^{4}-18 r^{2} s^{4}\)

Short Answer

Expert verified
The completely factored form is \(6r^2s^4(5q - 1)(2q + 3)\).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Examine each term in the expression to find the common factors. Each term has a common factor of \(6r^2s^4\). This factor is present because 60, 78, and 18 have a numerical GCF of 6, and all terms have at least \(r^2\) and \(s^4\) as factors. Thus, the GCF is \(6r^2s^4\).
02

Factor out the GCF

Divide each term of the expression by the GCF \(6r^2s^4\) and rewrite the expression. This simplifies to: \[60q^2r^2s^4 + 78qr^2s^4 - 18r^2s^4 = 6r^2s^4(10q^2 + 13q - 3)\]
03

Analyze the Quadratic Expression

The expression \(10q^2 + 13q - 3\) inside the parentheses needs to be examined to see if it can be factored further. Check for possible pairings that give a product of \(-30\) (since \(10\times -3 = -30\)) and a sum of \(13\).
04

Perform Factorization by Grouping

Identify two numbers that multiply to \(-30\) and add to \(13\). These are \(15\) and \(-2\). Use these to break down the middle term: \[10q^2 + 13q - 3 = 10q^2 + 15q - 2q - 3 \]. Regroup and factor by grouping: \[(10q^2 + 15q) + (-2q - 3) = 5q(2q + 3) - 1(2q + 3)\]. Factor out the common term: \[(5q - 1)(2q + 3)\].
05

Write the Entire Factored Expression

Combine all the factored parts together: \[60q^2r^2s^4 + 78qr^2s^4 - 18r^2s^4 = 6r^2s^4(5q - 1)(2q + 3)\].

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Key Concepts

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

Greatest Common Factor
The concept of the Greatest Common Factor (GCF) is a fundamental step in breaking down expressions into simpler components. The GCF is the largest expression that can be evenly divided into each term of a polynomial. In our example, to find the GCF, we start by looking at each term:
  • Numerical coefficients: 60, 78, and 18 share a greatest common factor of 6.
  • Variable components: Each term contains at least one factor of \(r^2\) and \(s^4\).
Combining these, the GCF for the entire expression is \(6r^2s^4\). This simplifies the expression significantly when factored out, paving the way for further simplification.
Quadratic Expressions
Quadratic expressions are polynomials of degree two. They are typically written in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In our original problem, once the GCF \(6r^2s^4\) is factored out, we are left with a quadratic expression \(10q^2 + 13q - 3\). Understanding how to further factor these expressions can simplify them more:
  • Identify the product of \(a\) and \(c\), which is \(-30\) (from \(10\times -3\)).
  • Find two numbers that multiply to \(-30\) and add to \(b = 13\). This would be 15 and -2.
These numbers help us split the middle term for further factorization, which is an essential technique for tackling quadratic expressions effectively.
Factorization by Grouping
Factorization by grouping involves rearranging a polynomial's terms so they can be grouped into factors that can be further simplified. This technique is particularly useful for quadratic expressions like \(10q^2 + 13q - 3\) from our example.To illustrate:
  • Start by rewriting the quadratic expression with the middle term split: \(10q^2 + 15q - 2q - 3\).
  • Next, group the terms: \((10q^2 + 15q) + (-2q - 3)\).
  • Factor each group separately: \(5q(2q + 3) - 1(2q + 3)\).
  • Notice \((2q + 3)\) is common, enabling further factorization: \((5q - 1)(2q + 3)\).
This final step gives us the factored form of the original quadratic expression, completing the process of factorization by grouping.

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