Problem 55 - Elementary Algebra

Question

Find the product. \((5 x+2)(5 x-2)\).

Step-by-Step Solution

Answer

Answer: The product of (5x+2)(5x-2) is 25x^2 - 4.

Step 1: First

Multiply the first terms of each binomial together: \((5x)(5x) = 25x^2\).

Step 2: Outer

Multiply the outer terms of each binomial together: \((5x)(-2) = -10x\).

Step 3: Inner

Multiply the inner terms of each binomial together: \((2)(5x) = 10x\).

Step 4: Last

Multiply the last terms of each binomial together: \((2)(-2) = -4\).

Step 5: Combine the terms

Combine the results from Steps 1-4 into a single expression: \(25x^2 - 10x + 10x - 4\).

Step 6: Simplify the expression

Simplify the expression by combining like terms: \(25x^2 - 10x + 10x - 4 = 25x^2 - 4\). The product of \((5x+2)(5x-2)\) is \(25x^2 - 4\).

Key Concepts

Binomials

When you hear the term "binomial," think of two terms put together. Binomials are a type of polynomial that specifically consist of two terms. For example,

• "5x + 2"
• "5x - 2"

Both are binomials because they each have two distinct terms.
In mathematics, binomials are often the ingredients we work with to form bigger algebraic expressions, through operations like addition, subtraction, and multiplication.
Recognizing a binomial is as simple as counting its terms. It is one of the basic building blocks in algebra.

Multiplication of Polynomials

When multiplying polynomials, like binomials, each term in one polynomial must be multiplied by every term in the other. For the binomials

• "(5x + 2)" and
• "(5x - 2)"

we follow a specific method known as FOIL, which stands for First, Outer, Inner, Last.
- **First**: Multiply the first terms, (e.g., \((5x) \times (5x) = 25x^2\)).
- **Outer**: Multiply the outer terms, (e.g., \((5x) \times (-2) = -10x\)).
- **Inner**: Multiply the inner terms, (e.g., \((2) \times (5x) = 10x\)).
- **Last**: Multiply the last terms, (e.g., \((2) \times (-2) = -4\)).
By multiplying these four pairs, we find the binomial product, before simplifying to arrive at the final expression.

Simplifying Expressions

Simplifying algebraic expressions helps in cleaning up and reducing them to their simplest forms. After multiplying polynomials, it's crucial to combine like terms.
Take the result from our previous step, which is \(25x^2 - 10x + 10x - 4\). Like terms are the terms that have the same variable raised to the same power. In our equation,

• "-10x" and
• "10x"

are like terms.
When combined, they cancel each other out because \(-10x + 10x = 0\).
Thus, the expression reduces to \(25x^2 - 4\). This is as simplified as it gets because there are no further like terms to combine. Simplifying allows us to understand expressions more clearly and prepares them for further use, such as solving equations or graphing.