Problem 90 - College Algebra with Modeling and Visualization

Question

Multiply the expressions. $$(x-7)^{2}$$

Step-by-Step Solution

Answer

The expanded expression is \(x^2 - 14x + 49\).

Step 1: Understand the Problem

We are asked to multiply the expression \((x-7)^2\). This means we need to expand it by multiplying \((x-7)\) by itself.

Step 2: Apply the Formula for Squaring a Binomial

Recall the formula for squaring a binomial: \((a-b)^2 = a^2 - 2ab + b^2\). Here, \(a = x\) and \(b = 7\).

Step 3: Square the First Term

Calculate \(a^2\). For \((x-7)^2\), the first term \((x)\) squared is \(x^2\).

Step 4: Calculate the Twice Product of the Two Terms

Calculate \(-2ab\). This equals to \(-2(x)(7) = -14x\).

Step 5: Square the Last Term

Calculate \(b^2\). For \(-7\), this is \((-7)^2 = 49\).

Step 6: Combine All Terms

Combine all the parts to get the expanded expression: \(x^2 - 14x + 49\).

Key Concepts

Squaring a Binomial

Squaring a binomial is a fundamental concept in algebra that many students encounter early on. When you square a binomial, you are essentially multiplying the binomial by itself. The formula to remember is

• \((a+b)^2 = a^2 + 2ab + b^2\)
• or \((a-b)^2 = a^2 - 2ab + b^2\)

This is how it works: take each term in the binomial, square them, and then multiply the two different terms together and double it. Finally, compile these results into a single expression.
For instance, consider the binomial \((x-7)^2\).

• The first term squared is \(x^2\),
• the twice product of the terms is \(-2 \times x \times 7 = -14x\),
• and the last term squared is \((-7)^2 = 49\).
• So overall, the expanded expression becomes \(x^2 - 14x + 49\).

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. For example,

• \(x^2 - 14x + 49\)

is an algebraic expression. These expressions can represent real-world quantities and are used extensively in equations and functions.
Algebraic expressions are crucial because they form the foundation for more complex mathematical concepts. They can be simplified, evaluated, or expanded, as in the exercise where you multiply \((x-7)(x-7)\). Being familiar with how to manipulate these expressions enables you to solve all kinds of algebraic problems.

• Addition and subtraction are used for combining like terms.
• Distribution and factoring help in simplifying or solving equations.

Learning how to effectively handle algebraic expressions paves the way for success in solving equations, inequalities, and much more.

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomials. It's essentially the process of distributing each term in the first polynomial over every term in the second polynomial. To make it more straightforward:

• Consider it as the distributive property being used over and over.

In our exercise \((x-7)(x-7)\),

• it's similar to multiplying two linear polynomials, and involves using the formula for squaring a binomial.
• First, you multiply each term in the first binomial by each term in the second binomial.
• This gives rise to multiple terms which are then combined together if they are like terms.
• Ultimately, this results in a single polynomial: \(x^2 - 14x + 49\).

Polynomial multiplication is a key skill in algebra, helping to simplify and solve various mathematical problems. Mastering it will provide a solid basis for understanding more advanced topics.