Problem 66 - Algebra A Combined Function

Question

Multiply. \(\left(m+\frac{2}{9}\right)\left(m-\frac{1}{9}\right)\)

Step-by-Step Solution

Answer

The expression simplifies to \( m^2 + \frac{m}{9} - \frac{2}{81} \).

Step 1: Identify the Expression

The given expression is \( \left(m + \frac{2}{9}\right)\left(m - \frac{1}{9}\right) \). This is a multiplication of two binomials.

Step 2: Use the FOIL Method

To expand the expression, use the FOIL method (First, Outer, Inner, Last). This technique helps in multiplying each term in the first binomial by each term in the second binomial.

Step 3: Multiply the First Terms

Multiply the first terms of each binomial: \( m \times m = m^2 \).

Step 4: Multiply Outer Terms

Multiply the outer terms from the two binomials: \( m \times \left(-\frac{1}{9}\right) = -\frac{m}{9} \).

Step 5: Multiply Inner Terms

Multiply the inner terms from the two binomials: \( \frac{2}{9} \times m = \frac{2m}{9} \).

Step 6: Multiply Last Terms

Multiply the last terms from each binomial: \( \frac{2}{9} \times \left(-\frac{1}{9}\right) = -\frac{2}{81} \).

Step 7: Combine Like Terms

Add all the products obtained from the previous steps: \( m^2 + \left(-\frac{m}{9}\right) + \frac{2m}{9} - \frac{2}{81} \).

Step 8: Simplify the Expression

Combine the like terms \( -\frac{m}{9} + \frac{2m}{9} \), which gives \( \frac{m}{9} \). The entire expression then simplifies to \( m^2 + \frac{m}{9} - \frac{2}{81} \).

Key Concepts

Understanding Binomials

Binomials are algebraic expressions with two terms. These terms are separated by either a plus or a minus sign. Each term may consist of variables, numbers, or a combination of both. In our given expression, the binomials are \(m + \frac{2}{9}\) and \(m - \frac{1}{9}\). These two binomials are grouped within parentheses, indicating that we must multiply them as part of the operation.
When working with binomials:

• Always identify the terms within the parentheses.
• Note the operation sign between the terms (either addition or subtraction).

Getting comfortable with binomials helps immensely when multiplying polynomials using the FOIL method.

Multiplying Polynomials with FOIL

Multiplying polynomials requires a systematic method, particularly with binomials. The FOIL method is a popular technique, often used in this circumstance. FOIL stands for First, Outer, Inner, and Last - these refer to the different combinations of terms to multiply from each binomial.
Here's how the FOIL method applies:

• First: Multiply the first terms in each binomial. For \((m + \frac{2}{9})(m - \frac{1}{9})\), this step gives \(m \times m = m^2\).
• Outer: Multiply the outer terms. This results in \(m \times -\frac{1}{9} = -\frac{m}{9}\).
• Inner: Multiply the inner terms. Similarly, \(\frac{2}{9} \times m = \frac{2m}{9}\).
• Last: Multiply the last terms in each binomial. Here, the product is \(\frac{2}{9} \times -\frac{1}{9} = -\frac{2}{81}\).

After completing these steps, combine all the resulting terms to simplify the expression.

Algebraic Expressions Simplification

After multiplying the polynomials using the FOIL method, it's critical to simplify the algebraic expression. Simplification involves combining like terms and reducing the expression to its simplest form.
In our example from \((m + \frac{2}{9})(m - \frac{1}{9})\), simplifying proceeds as follows:

• We begin with all products: \(m^2, -\frac{m}{9}, \frac{2m}{9}, -\frac{2}{81}\).
• Notice that \(-\frac{m}{9}\) and \(\frac{2m}{9}\) are like terms since both have the variable 'm' with the same denominator of 9.
• Combining these like terms results in \(\frac{(2m - m)}{9} = \frac{m}{9}\).
• Finally, write all simplified components together: \(m^2 + \frac{m}{9} - \frac{2}{81}\).

Thus, understanding binomials and using the FOIL method are foundational steps towards mastering the simplification of algebraic expressions when multiplying polynomials.