Problem 123 - Elementary Algebra

Question

Factor each expression. \(m^{2}+m n-56 n^{2}\)

Step-by-Step Solution

Answer

(m - 7n)(m + 8n)

Step 1 - Identify the quadratic expression

Recognize that the given expression is a quadratic trinomial: \(m^2 + mn - 56n^2\).

Step 2 - Find two numbers that multiply and add

Find two numbers that multiply to the constant term, \(-56n^2\), and add to the coefficient of the middle term, \(1n\). These numbers are \(8n\) and \(-7n\), because \(8n \times (-7n) = -56n^2\) and \(8n - 7n = n\).

Step 3 - Split the middle term

Use the two numbers found in Step 2 to split the middle term: \(m^2 + 8mn - 7mn - 56n^2\).

Step 4 - Factor by grouping

Group the terms and factor each group: \[(m^2 + 8mn) + (-7mn - 56n^2) \]Factor out the common terms: \[m(m + 8n) - 7n(m + 8n)\].

Step 5 - Factor the common binomial

Notice that \(m + 8n\) is common in both groups: \[(m - 7n)(m + 8n)\].

Key Concepts

Quadratic Trinomials

A quadratic trinomial is a polynomial with three terms. It has the general form:

• x^2 + bx + c, where 'a', 'b', and 'c' are constants.

In this definition, 'a' represents the coefficient of the quadratic term x^2, 'b' is the coefficient of the linear term 'x', and 'c' is the constant term. Here, the quadratic trinomial we worked with had 'm' and 'n' as variables. This means it follows the same principles:

• m^2 + mn - 56n^2

Recognizing quadratic trinomials allows us to use factoring techniques to simplify or solve them. Quadratics can generally be identified by their highest exponent of '2'; in this case, the highest exponent is on 'm' with m^2.

Factoring by Grouping

Factoring by grouping is a technique used when dealing with polynomials that have four or more terms. The idea is to group terms into pairs and then factor out a common factor from each pair. Let's see how it applies to our expression

• m^2 + 8mn - 7mn - 56n^2

First, group the terms:

• (m^2 + 8mn)
• (-7mn - 56n^2)

Then, factor out the common terms from each group. For the first group, the common term is 'm', giving us

• m(m + 8n)

For the second group, the common term is '-7n', giving us:

• -7n(m + 8n)

So our expression looks like this:

• m(m + 8n) - 7n(m + 8n)

Notice that (m + 8n) is common in both terms.

Splitting the Middle Term

Splitting the middle term is a method used to factor quadratic trinomials. It involves finding two numbers that multiply to the product of the constant term and the coefficient of the quadratic term. These numbers should also add up to the coefficient of the middle term. In our example:

• Expression: m^2 + mn - 56n^2

We need to find two numbers whose product is

• The constant term: -56n^2
and whose sum is
• The coefficient of the middle term: 1n
The numbers that work are 8n and -7n. We use these to split the middle term 'mn':
• m^2 + 8mn - 7mn - 56n^2
• By doing this, we've rewritten the trinomial as a four-term polynomial, which can be factored by grouping. This method is particularly useful when straightforward factoring isn't obvious.