Problem 50 - Introductory Algebra for College Students

Question

Factor any perfect square trinomials, or state that the polynomial is prime. $$x^{2}-4 x+4$$

Step-by-Step Solution

Answer

The factored form of the perfect square trinomial \(x^{2}-4 x+4\) is \((x - 2)^2\).

Step 1: Identify 'a' and 'b'

In the given trinomial, \(x^{2} - 4x + 4\), 'a' is the square root of the first term and 'b' is the square root of the third term. Therefore, \(a = x\) and \(b = 2\).

Step 2: Check if it is a Perfect Square Trinomial

For the trinomial to be a perfect square trinomial, the second term should be twice the product of 'a' and 'b'. In this case, the second term is '-4x', and twice the product of 'a' and 'b' is also '-4x'. Therefore, \(x^{2} - 4x + 4\) is a perfect square trinomial.

Step 3: Factor the Perfect Square Trinomial

Since it's a perfect square trinomial, the factored form will be \((a - b)^2\). Substituting 'a' and 'b' gives \((x - 2)^2\).

Key Concepts

Factoring Polynomials

Factoring polynomials is like solving a puzzle. You get a polynomial and your task is to break it down into simpler pieces, similar to how you put together smaller parts to complete a picture. When a polynomial is factored completely, it cannot be broken down any further using integer coefficients. One special type is the perfect square trinomial. This is a polynomial that can be rewritten as a single binomial squared.

In the example provided, the trinomial was factored as \[(x - 2)^2\].

• The first term, \(x^2\), appeared because \(x\) was squared.
• The last term, \(4\), came from squaring \(2\).
• The middle term, \(-4x\), is the result of multiplying 'a', 'b', and 2 (from \(2ab\)).

By identifying the value of 'a' and 'b', and verifying the middle term matches the requirement of a perfect square trinomial, you end up with a neatly factored expression like \[(x - 2)^2\]. This represents the solution to the factoring puzzle!

Algebraic Expressions

Algebraic expressions are like math sentences that include numbers, variables, and operators (like plus, minus, times). They help us describe relationships and solve problems. A trinomial, which is the type of expression used in the exercise, is an algebraic expression composed of three terms. For example, in \(x^2 - 4x + 4\), there's a term with a square, a linear term, and a constant.

When dealing with perfect square trinomials, the structure becomes predictable because it follows a specific formula:

• The first term is a square, e.g., \(x^2\).
• The third term is also a square, e.g., \(4\).
• The middle term is twice the product of the bases of the squared terms, \(-4x\).

Understanding how these parts fit together helps you see why the trinomial can be broken down to \[(x - 2)^2\]. It's like recognizing the components of a sentence, allowing you to restructure it in its simplest form.

Introductory Algebra

Introductory algebra introduces basic concepts that are used throughout math. It builds upon arithmetic by introducing components like variables and introduces techniques for solving equations and simplifying expressions. One key concept is understanding how to manipulate algebraic expressions; including recognizing and factoring trinomials.

In the example provided, recognizing \(x^2 - 4x + 4\)as a perfect square trinomial is fundamental. The process involves:

• Identifying if the structure matches the pattern \((a - b)^2 = a^2 - 2ab + b^2\).
• Verifying terms by checking whether the second term equals twice the product of the roots of the first and third terms.
• Factoring the expression accurately to simplify it into an easily understandable form.

As beginners in algebra, practicing these steps enhances problem-solving skills and reinforces an understanding of polynomial structures, making algebra a less daunting subject.