Problem 22 - 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. $$(x-2)^{2}$$

Step-by-Step Solution

Answer

\(x^2 - 4x + 4\)

Step 1: Identify the Expression

The given expression is \((x-2)^2\)This means we need to multiply \((x-2)\) by itself.

Step 2: Expand the Expression

We will use the formula for the square of a binomial: \((a - b)^2 = a^2 - 2ab + b^2\)Here, \(a = x\) and \(b = 2\).

Step 3: Apply the Formula

Substitute the values of \(a\) and \(b\) into the formula:\(x^2 - 2 \times x \times 2 + 2^2\).

Step 4: Simplify the Terms

Calculate each term:- The first term, \(x^2\), stays the same.- The second term, \(-2 \times x \times 2\), simplifies to \(-4x\).- The third term, \(2^2\), simplifies to 4.

Step 5: Write the Final Expression

Combine the simplified terms:\(x^2 - 4x + 4\).This is the expanded form of the expression \((x-2)^2\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a foundational concept in algebra that provides a formula for expanding powers of a binomial. A binomial is simply an algebraic expression containing two terms, like \( (x + y) \). The theorem is particularly useful because it allows us to expand expressions such as \( (x - 2)^2 \) without directly multiplying the terms.

The general formula for the nth power of a binomial \( (a + b)^n \) is:

• \( \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \)

This formula uses coefficients that are determined by the binomial coefficients, denoted by the symbol \( \binom{n}{k} \), which are the numbers from Pascal's Triangle.

In our exercise with \( (x - 2)^2 \), the binomial theorem simplifies to a formula for the square of a binomial: \( (a - b)^2 = a^2 - 2ab + b^2 \). This gives us a straightforward way to expand and simplify binomials raised to a power.

Polynomial Expansion

Polynomial expansion refers to the process of expressing a power of a binomial as a sum of terms, or in other words, expanding it into a polynomial.

Take for example \( (x - 2)^2 \). To expand this expression, we multiply each part of the binomial:

• First term: \( x^2 \)
• Second term: \(-2 \times x \times 2 = -4x \)
• Third term: \( 2^2 = 4 \)

When combined, they create the expanded polynomial \( x^2 - 4x + 4 \).

By expanding binomials, we turn them into polynomials that are easier to work with in computations and analysis. This makes polynomial expansion an indispensable tool in algebra and calculus.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and mathematical operations. They form the building blocks of algebra and appear in countless mathematical problems and equations. Understanding how to manipulate and expand these expressions is crucial.

Expressions like \( (x - 2)^2 \) are specific types of algebraic expressions known as binomials. Expanding them involves rewriting them in a standard polynomial form. For example:

• An algebraic expression could be \( x^2 - 4x + 4 \)

It's a polynomial, and understanding how to switch between these forms is an essential skill.

Algebraic expressions can represent real-world quantities, from area and volume to speed and distance. Practicing with these expressions hones skills in pattern recognition, problem solving, and logical thinking.