Problem 64 - Algebra A Combined Function

Question

Mixed Practice Multiply. $$ (x-10)(x+10) $$

Step-by-Step Solution

Answer

The expanded form is \(x^2 - 100\).

Step 1: Recognize the Pattern

The expression \((x - 10)(x + 10)\) follows the special product pattern known as the difference of squares.

Step 2: Apply the Difference of Squares Formula

The difference of squares formula is \((a - b)(a + b) = a^2 - b^2\). In your case, identify \(a = x\) and \(b = 10\).

Step 3: Substitute into the Formula

Substituting into the formula gives \(x^2 - 10^2\).

Step 4: Simplify the Expression

Calculate \(10^2\), which is 100, so the expression becomes \(x^2 - 100\).

Key Concepts

Difference of Squares

In algebra, the difference of squares is a helpful tool for simplifying certain expressions. It's recognized when you have two terms, both being perfect squares, separated by a subtraction sign. The structure

• One squared term minus another squared term
• Expressed as \(a^2 - b^2\)

This pattern allows us to quickly factor the expression into a product of two binomials. When you see something like

• \((x - 10)(x + 10)\)

it fits perfectly into this category because of its resemblance to

• \((a - b)(a + b)\)

which equals

• \(a^2 - b^2\)

Applying this understanding simplifies expressions greatly.

Special Product Patterns

Special product patterns are shortcuts in algebra that help in multiplying expressions more efficiently. They include formulas like

• the difference of squares
• the square of a binomial
• the sum and difference of cubes

In the given example

• \((x - 10)(x + 10)\)

it demonstrates the special product pattern of the difference of squares. Using this pattern, you can directly obtain \(x^2 - 100\) from

• \((x - 10)(x + 10)\)

without expanding the terms individually. This saves both time and effort and reduces the chances of errors in calculations.

Factoring Expressions

Factoring is breaking down an expression into a product of simpler components. In algebra, it’s a necessary skill for simplifying expressions and solving equations. Here’s how you do it:

• Identify a pattern—in this case, a difference of squares.
• Substitute the terms into the pattern; \((x - 10)(x + 10)\) becomes \(x^2 - 10^2\).
• Simplify by calculating, \(10^2 = 100\), resulting in the expression \(x^2 - 100\).

Factoring is crucial for mathematical problem-solving and aids in understanding the relationships between numbers and expressions. Once you learn to spot these patterns, working with polynomials becomes much more straightforward.