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Chapter 10: Problem 25

Multiply the polynomials. $$(m-4)(m+7)$$

Short Answer

Expert verified
The product is \ m^2 + 3m - 28 \.

Step by step solution

01

Apply the Distributive Property

To multiply the polynomials \(m-4\) and \(m+7\), apply the distributive property (also known as the FOIL method for binomials): \[ \ (m-4)(m+7) = m(m+7) - 4(m+7) \]
02

Distribute Each Term

Distribute each term separately: \[ \ m(m+7) = m \cdot m + m \cdot 7 = m^2 + 7m \] \[ -4(m+7) = -4 \cdot m + (-4) \cdot 7 = -4m - 28 \]
03

Combine Like Terms

Combine the like terms from the expressions: \[ \ m^2 + 7m - 4m - 28 \] Simplify by adding the coefficients of the like terms: \[ \ m^2 + 3m - 28 \]

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Key Concepts

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

distributive property
The distributive property is a fundamental concept in algebra. It allows us to multiply a sum or difference by distributing the multiplication across each addend or subtrahend. When multiplying polynomials, the distributive property helps break down the problem into simpler parts.
For example, with the polynomial multiplication \( (m - 4)(m + 7) \), we use the distributive property to expand it:
  • First, distribute \( m \) across \( m + 7 \)
  • Then, distribute \( -4 \) across \( m + 7 \)
This results in:
\ m(m + 7) - 4(m + 7) \
Each term is multiplied separately and then combined.
combining like terms
Combining like terms simplifies the polynomial by merging terms that have the same variables to the same power. This step is crucial because it helps in reducing the polynomial to its simplest form.
Consider the expanded polynomial from the previous step:
\ m^2 + 7m - 4m - 28 \
Here, \( 7m \) and \( -4m \) are like terms because they both contain the variable \( m \). To combine them, we simply add their coefficients:
  • 7m - 4m = 3m
So, the simplified polynomial becomes:
\ m^2 + 3m - 28 \
Always look for like terms and combine them to simplify your polynomial expressions.
FOIL method
The FOIL method is a special case of the distributive property used specifically for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, representing the pairs of terms you multiply together.
Given the example \( (m - 4)(m + 7) \), let's apply the FOIL method:
  • First: Multiply the first terms of each binomial: \( m \times m = m^2 \).
  • Outer: Multiply the outer terms: \( m \times 7 = 7m \).
  • Inner: Multiply the inner terms: \( -4 \times m = -4m \).
  • Last: Multiply the last terms: \( -4 \times 7 = -28 \).
When we combine these results, we get:
\ m^2 + 7m - 4m - 28 \
By following these steps, you ensure that all terms are correctly multiplied and accounted for in polynomial multiplication.

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