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Chapter 5: Problem 24

Multiply. $$(-m+5)(2 m-9)$$

Short Answer

Expert verified
-2m^2 + 19m - 45

Step by step solution

01

- Apply the Distributive Property

Use the distributive property, also known as FOIL (First, Outer, Inner, Last), to multiply each term in the first binomial by each term in the second binomial. This gives: (-m + 5)(2m - 9) = (-m)(2m) + (-m)(-9) + 5(2m) + 5(-9)
02

- Multiply the Individual Terms

Calculate the product of each pair of terms. This gives: -m * 2m = -2m^2 -m * -9 = 9m 5 * 2m = 10m 5 * -9 = -45
03

- Combine Like Terms

Add the products together and combine like terms. This gives: -2m^2 + 9m + 10m - 45 Combine the like terms 9m and 10m: -2m^2 + 19m - 45

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Key Concepts

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

Binomial Multiplication
Let's begin with understanding the multiplication of binomials. Binomials are algebraic expressions that contain exactly two terms. The given problem is to multiply two binomials: (-m + 5) and (2m - 9). This means we need to find the product of these expressions. To do this, we use a method that involves distributing every term in the first binomial to every term in the second binomial. This might sound complex, but it becomes straightforward with practice. Remember:
Distributing here means we are going to multiply each term in the first binomial by each term in the second.
Like Terms
Next, let's talk about like terms. When we multiply and expand binomials, we often end up with terms that look similar. In algebra, we call these like terms. Like terms are terms that have the same variables raised to the same power, even though they might have different coefficients.

For example, in the problem: after distributing, we get the terms:
-2m^2, 9m, 10m, and -45.
* -2m^2 is its own term because it contains m^2.
* 9m and 10m are like terms because they both contain m.
* -45 is a constant term and stands alone.
When you identify like terms, you can combine them by adding their coefficients. Thus, we add 9m and 10m together to get 19m.
FOIL Method
The FOIL method is a specific technique used to multiply binomials. FOIL stands for First, Outer, Inner, Last. This acronym refers to the position of the terms in the binomials that should be multiplied. Let's break it down with our given example:
  • First: Multiply the first terms in each binomial. Here, it's (-m) and (2m), giving us -2m^2.
  • Outer: Multiply the outer terms in the product. These are (-m) and (-9), giving us +9m.
  • Inner: Multiply the inner terms. Here, it's (+5) and (2m), producing +10m.
  • Last: Multiply the last terms in each binomial. These are (+5) and (-9), giving us -45.

By using FOIL, you can effectively ensure you've accounted for all parts of the binomial multiplication. Once you have all the products, combining like terms (as previously discussed) gives the final simplified expression:
-2m^2 + 19m - 45.

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