Problem 8 - Algebra 1: Concepts and Skills

Question

Use a special product pattern to find the product. $$ (a-2)(a+2) $$

Step-by-Step Solution

Answer

The product is \(a^2 - 4\).

Step 1: Identify the pattern

In this case, we see that we have the formula of the difference of two squares. This pattern can be written as \( (a+b)(a-b) \) or \( (a-b)(a+b) \). The terms \( a-2 \) and \( a+2 \) fit this pattern, with \( a \) as \( a \) and \( 2 \) as \( b \).

Step 2: Apply the special product pattern

The pattern \((a+b)(a-b) = a^2 - b^2\) tells us that we can subtract the square of the second term from the square of the first term to get the product. So, in our expression \((a-2)(a+2)\), we consider \(a\) to be our first term and \(2\) to be the second term.

Step 3: Calculate the product

Substitute \(a\) and \(2\) into the difference of squares formula: \(a^2 - 2^2\).

Key Concepts

Special Product Patterns

When dealing with algebraic expressions, recognizing special product patterns can simplify your calculations. One common pattern is the "difference of squares." This occurs when you have two binomials that are identical except for their signs, such as

• \((a-b)(a+b)\)
• \((a+b)(a-b)\)

In this pattern, the "difference" refers to the subtraction in the binomials, and the "squares" refer to squaring the terms inside each binomial. The formula tells us that we can multiply these binomials by simply subtracting the square of the second term from the square of the first term: \[a^2 - b^2\]This method avoids tedious distribution, making it faster.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. They form the building blocks of algebra, acting as a sort of mathematical shorthand for various computations.In an expression like

• \(a-2\)
• \(a+2\)

you see both a variable \(a\) and a constant \(2\). Variables represent unknown or changeable numbers, while constants hold set values. Manipulating these expressions often involves combining like terms or applying special patterns, like the difference of squares, to make calculations simpler and expressions more manageable.

Multiplying Binomials

Multiplying binomials is a key skill in algebra. Using traditional methods like the distributive property, you would multiply each term in the first binomial with each term in the second.However, recognizing special patterns, like the difference of squares, can simplify this process substantially. With two binomials like

• \((a-2)\)
• \((a+2)\)

you can apply the difference of squares formula: \[(a+b)(a-b) = a^2 - b^2\]This approach results in fewer steps than using the distributive property, reducing the likelihood of errors and speeding up the calculation. By mastering this skill, you can quickly work with and simplify polynomial expressions.