Problem 56 - Elementary Algebra

Question

For the following problems, find the products. $$ (5 x+6)(5 x-6) $$

Step-by-Step Solution

Answer

Answer: The product of the given binomials (5x + 6)(5x - 6) is 25x^2 - 36.

Step 1: Find the product of the First terms

To find the product of the first terms in each binomial, multiply \((5x)\) by \((5x)\). This gives us: $$ (5x)(5x) = 25x^2 $$

Step 2: Find the product of the Outer terms

Next, we will find the product of the outer terms. Multiply \((5x)\) by \((-6)\). This gives us: $$ (5x)(-6) = -30x $$

Step 3: Find the product of the Inner terms

Now, let's find the product of the inner terms. Multiply \((6)\) by \((5x)\). This gives us: $$ (6)(5x) = 30x $$

Step 4: Find the product of the Last terms

Finally, we need to find the product of the last terms in each binomial. Multiply \((6)\) by \((-6)\). This gives us: $$ (6)(-6) = -36 $$

Step 5: Combine like terms

Now, we will combine the terms obtained in steps 1 to 4: $$ 25x^2 - 30x + 30x - 36 $$ Notice that the second and third terms, \(-30x\) and \(+30x\), are like terms and cancel each other out. So, the final product is: $$ 25x^2 - 36 $$ Therefore, the product of the given binomials \((5x + 6)(5x - 6)\) is \(25x^2 - 36\).

Key Concepts

Binomials

A binomial is an expression that consists of two distinct terms connected by a plus or minus sign. For instance, in the exercise given, we have the binomials

• \((5x + 6)\)
• \((5x - 6)\)

Binomials can represent many different scenarios, but they often appear in algebraic expressions where we may need to perform operations like addition, subtraction, or multiplication.
The name 'binomial' comes from Latin, where "bi-" means two and "-nomial" refers to terms, so essentially, it's a two-termed expression.
Understanding how to work with binomials is crucial, as they are foundational elements in algebra.

Distribution

The process of polynomial multiplication often involves something called distribution. This refers to the technique where each term in one polynomial is multiplied by each term in the other polynomial.
For the problem at hand, this means applying the distribution property to multiply each term in the binomial

• \((5x + 6)\)
• \((5x - 6)\)

The Distributive Property is symbolized by a(b + c) = ab + ac. In our case, each part of the binomial is distributed over the other binomial’s terms.
A common method to organize this is the FOIL method, which helps us remember the sequence to multiply in: First, Outer, Inner, and Last terms.

Combine Like Terms

Once each pair of terms has been distributed and multiplied, the collected terms often need to be simplified by combining like terms. Like terms are terms that have the same variables raised to the same power.
After completing the distribution steps in the binomial multiplication, we obtained:\[25x^2 - 30x + 30x - 36\]
The terms

• \(-30x\)
• \(+30x\)

are like terms since they both have the variable x raised to the power of one. In this case, the like terms cancel each other out.
Consequently, the expression simplifies to \(25x^2 - 36\), which is the final result after combining like terms. Being able to spot and combine like terms is a key skill when working with any polynomial expression to ensure clarity and correctness in your solutions.