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Chapter 6: Problem 141

In the following exercises, factor each expression using any method. $$ 6 n^{2}+5 n-4 $$

Short Answer

Expert verified
\((3n + 4)(2n - 1)\)

Step by step solution

01

Identify the quadratic expression

The given expression is a quadratic polynomial of the form \(6n^{2} + 5n - 4\).
02

Find two numbers that multiply to the product of the coefficient of \(n^2\) and the constant term

Multiply the coefficient of \(n^2\) (which is 6) by the constant term (which is -4). This product is -24. Look for two numbers that multiply to -24 and add up to the coefficient of \(n\), which is 5. These numbers are 8 and -3.
03

Rewrite the middle term using the two numbers found

Rewrite the quadratic polynomial by splitting the middle term using the numbers found in Step 2: \(6n^2 + 8n - 3n - 4\)
04

Factor by grouping

Group the terms: \((6n^2 + 8n) + (-3n - 4)\) Factor out the greatest common factor (GCF) from each group: \(2n(3n + 4) - 1(3n + 4)\)
05

Factor out the common binomial factor

Factor out the common binomial \((3n + 4)\): \((3n + 4)(2n - 1)\)

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Key Concepts

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

Quadratic Polynomial
A quadratic polynomial is a type of polynomial where the highest degree of any term is 2. These expressions are generally written in the standard form: \(ax^2 + bx + c\). Here, \(a\), \(b\), and \(c\) are constants, with \(a\) being non-zero.
Factoring by Grouping
Factoring by grouping is a method used to factor polynomials, especially when the polynomial has four terms. It involves breaking up the polynomial into groups, factoring out the greatest common factor in each group, and then factoring out the common binomial factor. For example, in the expression \(6n^2 + 5n - 4\), we rewrite the middle term to form two groups: \((6n^2 + 8n) + (-3n - 4)\). Then, we factor each group separately.
Greatest Common Factor (GCF)
The Greatest Common Factor (GCF) is the largest factor that can be evenly divided into each term in a polynomial. Finding the GCF is a crucial step in factoring by grouping. For example, in \((6n^2 + 8n)\), the GCF is \(2n\) and for \((-3n - 4)\), the GCF is \(-1\). Factoring out these GCFs simplifies the expression and prepares it for the next step.
Binomial Factors
Binomial factors are expressions containing two terms, usually in the form \((ax + b)\). After grouping and factoring out the GCF from each set, the next step involves identifying and factoring out the common binomial factor. In the example problem, after factoring by grouping, our expression becomes \(2n(3n + 4) - 1(3n + 4)\). The common binomial factor here is \((3n + 4)\), leading to the final factored form: \((3n + 4)(2n - 1)\).

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Most popular questions from this chapter

In the following exercises, factor each trinomial of the form \(x^{2}+b x y+c y^{2} .\) $$ m^{2}-5 m n+30 n^{2} $$

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In the following exercises, factor each expression using any method. $$ 12 x^{2}+36 y-24 z $$

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In the following exercises, foctor using substitution. $$ (x-3)^{2}-5(x-3)-36 $$
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