Problem 51 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(5 x-2 a)(5 x+2 a)$$

Step-by-Step Solution

Answer

The product is \(25x^2 - 4a^2\).

Step 1: Identify the Binomial Pattern

We are given the expression \((5x - 2a)(5x + 2a)\). This fits the pattern of the product of a sum and a difference, also known as the difference of squares: \((A + B)(A - B) = A^2 - B^2\). Here, \(A = 5x\) and \(B = 2a\).

Step 2: Apply the Difference of Squares Formula

Using the identified pattern, substituting \(A = 5x\) and \(B = 2a\) into the formula, we calculate:\[(5x)^2 - (2a)^2\]

Step 3: Calculate Each Square

First, compute \((5x)^2\):\((5x)^2 = 25x^2\).Next, compute \((2a)^2\):\((2a)^2 = 4a^2\).

Step 4: Subtract the Squares

Substitute the squared terms back into the difference of squares formula:\[25x^2 - 4a^2\]

Step 5: Write the Final Expression

We now have the final product: \[25x^2 - 4a^2\]. This is the result of multiplying \((5x - 2a)(5x + 2a)\).

Key Concepts

Difference of Squares

The difference of squares is a special algebraic identity that allows us to simplify certain expressions easily. It involves the product of two binomials, where one is a sum and the other a difference of the same terms. In other words, we look at expressions like

• \((A + B)(A - B)\), which simplifies to \(A^2 - B^2\).

This identity is powerful because it reduces the need for direct multiplication, making calculations quicker. In our case, by recognizing

• \((5x - 2a)(5x + 2a)\) as following this pattern, we can directly write the result as \(25x^2 - 4a^2\).

To apply this, you simply square each term separately, as shown. Remember, this pattern only works when you have the precise setup of a sum and a difference, each sharing the same terms.

Binomials

A binomial is an algebraic expression with two terms. For example,

• \(5x - 2a\) and \(5x + 2a\) are both binomials.

Binomials are common in algebra, often involved in operations such as addition, subtraction, and multiplication.
Understanding the structure of a binomial is crucial when using patterns like the difference of squares. A binomial typically looks like

• \(A + B\) or \(A - B\).

These two forms form the basis for several algebraic identities, including binomial expansion and factoring.

Multiplying Polynomials

Multiplying polynomials involves performing the distributive property, where every term in one polynomial multiplies every term in the other. When two binomials are multiplied together, you follow the rule of distributing each term. For instance,

• the expression \((5x - 2a)(5x + 2a)\) could be expanded by multiplying each term in the first binomial by each term in the second.
• That process yields the result, which we directly simplified using the difference of squares to \(25x^2 - 4a^2\).

In general, when multiplying any polynomials, pay attention to combining like terms and following the order of operations carefully. For larger polynomials, understanding special patterns, like those for binomial products, can greatly simplify the process.