Problem 35 - Algebra and Trigonometry

Question

In Exercises 15–58, find each product. $$ (5-7 x)(5+7 x) $$

Step-by-Step Solution

Answer

\[25 - 70x - 49x^2\]

Step 1: First Terms Multiplication

The first terms are 5 from each of the binomials. When multiplied together, 5 times 5 gives 25.

Step 2: Outer Terms Multiplication

The outer terms are 5 and -7x. When multiplied together, 5 times -7x gives -35x.

Step 3: Inner Terms Multiplication

The inner terms are -7x and 5. When multiplied together, -7x times 5 gives -35x.

Step 4: Last Terms Multiplication

The last terms are -7x from the first binomial and 7x from the second. When multiplied together, -7x times 7x gives -49x².

Step 5: Combine Like Terms

Combining like terms, we add together the products of the outer and inner terms, then add the rest. This gives 25 - 35x - 35x - 49x². Simplifying further, this becomes \[25 - 70x - 49x^2\].

Key Concepts

Binomials

Binomials are expressions with exactly two terms. These terms are typically connected by either an addition or subtraction operation. In our example, the binomials \((5 - 7x)\) and \((5 + 7x)\) have each their own unique characteristics. The terms in the first binomial are 5 and \(-7x\), while the second binomial has terms 5 and \(7x\). Understanding that each term in a binomial can represent different values is key to effectively multiplying binomials together.
Couple of things to remember when working with binomials:

• Each term in the binomial can stand alone in a mathematical operation.
• Binomials can be multiplied, just like single numbers, often using specific methods like the FOIL Method to find their products.

Spotting binomials and their unique components is the first step in multiplying them correctly.

FOIL Method

The FOIL method is a technique commonly used for multiplying two binomials. It's an acronym that stands for First, Outer, Inner, Last, which refers to the order in which you multiply the terms. Let’s break it down using the binomials \((5 - 7x)\) and \((5 + 7x)\):

• First: Multiply the first terms in each binomial, here it’s 5 and 5, resulting in 25.
• Outer: Multiply the outer terms, 5 and \(7x\), to get \(-35x\).
• Inner: Multiply the inner terms, \(-7x\) and 5, which gives \(-35x\).
• Last: Multiply the last terms, \(-7x\) and \(7x\), concluding in \(-49x^2\).

Once all individual product calculations are complete, the results are combined to give the final expanded expression. The FOIL method is efficient for binomial multiplication, keeping track of contributions from each interaction between the terms.

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term across each term within a binomial or larger expression. In the context of our example, the property ensures every term in the first binomial is multiplied by every term in the second binomial.
For \((5 - 7x)(5 + 7x)\), the distributive property is working when you:

• Multiply 5 by each term in \((5 + 7x)\)
• Then multiply \(-7x\) by each term in \((5 + 7x)\)

It ensures no term is overlooked and every interaction is calculated. Think of it as spreading or distributing each term across the other binomial to expand them fully.

Like Terms

Combining like terms is an essential step in simplifying polynomial expressions after using methods like FOIL or Distributive Property. Like terms in mathematics are terms that have identical variable parts raised to the same power. They are the terms you can directly add or subtract from each other.
In our example, after using the FOIL method, the products of the outer and inner terms \(-35x\) are like terms. Here's how you combine them:

• Add \(-35x\) to \(-35x\), which simplifies to \(-70x\).

The expression ultimately simplifies to \(25 - 70x - 49x^2\), showcasing how like terms help reduce complexity and make understanding algebraic expressions easier.