Problem 32 - College Algebra Essentials

Question

Find each product. $$(x+5)(x-5)$$

Step-by-Step Solution

Answer

The product of the binomials \( (x+5)(x-5) \) is \( x^2 - 25 \)

Step 1: Apply the Distributive Property

Begin by applying the distributive property, which states that multiplication distributes over addition and subtraction. This means that each term inside the first binomial \((x+5)\) must be multiplied by each term inside the second binomial \((x-5)\). Thus, we get four products initially: \(x * x, x*(-5), 5*x\) and \(5*(-5)\).

Step 2: Simplify the Product

Simplify the expressions obtained from the distribution. The multiplication of \(x*x\) equals \(x^2\), \(x*(-5)\) equals \(-5x\), \(5*x\) equals \(5x\), and \(5*(-5)\) equals \(-25\). Now, our equation looks like this: \(x^2 - 5x + 5x - 25\)

Step 3: Combine like terms

Combine all similar terms for final simplification. Notice that \(-5x\) and \(5x\) are like terms and can be added together. These terms will cancel out because they sum up to zero. Therefore, the simplified expression is \(x^2 - 25\)