Problem 15 - Elementary Algebra

Question

Find the following products and simplify. $$ (m-9)(m-2) $$

Step-by-Step Solution

Answer

Answer: The product of (m - 9)(m - 2) is m^2 - 11m + 18.

Step 1: Multiply the first terms

Multiply the first terms in each binomial, which are m and m, to get: $$ m \times m = m^2 $$

Step 2: Multiply the outer terms

Multiply the outer terms in each binomial, which are m and -2, to get: $$ m \times (-2) = -2m $$

Step 3: Multiply the inner terms

Multiply the inner terms in each binomial, which are -9 and m, to get: $$ (-9) \times m = -9m $$

Step 4: Multiply the last terms

Multiply the last terms in each binomial, which are -9 and -2, to get: $$ (-9) \times (-2) = 18 $$

Step 5: Combine all the terms

Now, combine all the terms from Steps 1 to 4 to get the simplified product: $$ m^2 - 2m - 9m + 18 $$

Step 6: Combine like terms

Combine the like terms in the expression, which are the -2m and -9m terms, to get the final simplified product: $$ m^2 - 11m + 18 $$ So the product of (m - 9)(m - 2) is: $$ (m - 9)(m - 2) = m^2 - 11m + 18 $$

Key Concepts

Understanding Binomials

A binomial is a type of algebraic expression that contains exactly two terms. These terms are usually separated by a plus or minus sign. For example, in the expression \( (m - 9) \), we have two terms:

• \( m \) - the first term, and
• -9 - the second term.

In algebra, understanding binomials is crucial for various operations, such as addition, subtraction, and particularly multiplication like in this exercise. Binomials can be multiplied using several methods, and one popular method is the FOIL method, which is especially effective when dealing with two binomials.

Exploring the FOIL Method

The FOIL method is a handy tool for multiplying two binomials. FOIL stands for:

• First
• Outer
• Inner
• Last

This acronym helps organize the multiplication process. To use it, we multiply each term in the first binomial by each term in the second binomial, in a specific order:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outermost terms in the product.
• Inner: Multiply the innermost terms.
• Last: Multiply the last terms of each binomial.

By applying FOIL, you ensure that all necessary products are found. After multiplying, all you have to do is combine these results, which often leads to identifying common and like terms in the expression.

Combining Like Terms

Once you have applied the FOIL method, you're often left with an expression consisting of several terms. Like terms are terms that have the same variables raised to the same power. In our example exercise, after using the FOIL method on \( (m - 9)(m - 2) \), we get \( m^2 - 2m - 9m + 18 \). Here, the like terms are -2m and -9m because both have the same variable, \( m \), with the same exponent. To combine them, simply add their coefficients together:\(-2m - 9m = -11m\).Combining like terms simplifies the expression and often results in a polynomial in its simplest form, which is essential for solving equations and further algebraic manipulations.

Simplification in Algebra

Simplification in algebra is about making an expression as concise as possible while retaining its value. After using binomial multiplication methods like FOIL and combining like terms, simplification is usually the final step. In our exercise, after performing multiplication and combining like terms, we end up with \( m^2 - 11m + 18 \).
Why simplify? Simplifying expressions:

• Makes calculations more straightforward and manageable.
• Helps quickly identify solutions to equations.
• Allows a clearer understanding of the relationships between variables in an equation.

By mastering simplification, you'll be able to work more efficiently with not only algebra but higher-level math as well. Remember, the simpler the expression, the easier it is to work with in algebraic equations.