Problem 120 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. All perfect square trinomials are squares of binomials.

Step-by-Step Solution

Answer

The statement is true, all perfect square trinomials are squares of binomials.

Step 1: Understanding the terms

Firstly, understanding what a perfect square trinomial (PST) and a square of a binomial are is crucial. A PST is a type of trinomial which looks like this: \(a^{2}+2ab+b^{2}\) or \(a^{2}-2ab+b^{2}\). The term 'perfect square' comes from the fact that it is produced when a binomial is squared. In contrast, a square of a binomial is the result when a binomial is multiplied by itself, i.e., \((a+b)^{2}\) or \((a-b)^{2}\). Thus, both definitions exhibit similar patterns.

Step 2: Testing the statement

As per the definitions, a perfect square trinomial is indeed the result of squaring a binomial. For instance, \((a+b)^{2}\) yields \(a^{2}+2ab+b^{2}\), which is a PST. Similarly, \((a-b)^{2}\) yields \(a^{2}-2ab+b^{2}\), again a PST. This implies that the original statement that all PSTs are squares of binomials is true.

Step 3: Concluding Verification

Based on the observed patterns and working out the squares of binomials, it confirms that the original statement is indeed correct. Therefore, all PSTs are, in fact, squares of binomials.

Key Concepts

Squares of Binomials

When we talk about squares of binomials, we're referring to a specific mathematical process. Squaring a binomial involves taking a binomial expression—like \(a+b\) or \(a-b\)—and multiplying it by itself. This process follows the algebraic principle that each term within the binomials must be multiplied by every other term.

So, if we have the binomial \(a+b\), squaring it looks like this: \(a+b)^{2} = (a+b)(a+b)\). When we perform the multiplication, we apply the distributive property, also known as the FOIL method (First, Outside, Inside, Last), to get \(a^2 + ab + ab + b^2\), which simplifies to \(a^2 + 2ab + b^2\).

Similarly, squaring \(a-b\) follows the same logic: \(a-b)^{2} = (a-b)(a-b)\), producing \(a^2 - ab - ab + b^2\), which simplifies to \(a^2 - 2ab + b^2\). These resulting expressions, \(a^2 + 2ab + b^2\) and \(a^2 - 2ab + b^2\), are perfect square trinomials because they can be factored back into their original binomial forms.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations. Expressions can vary in complexity from simple forms like \(2x+3\) to more complex arrangements involving exponents, like the perfect square trinomials we discuss here.

Perfect square trinomials are a particular type of algebraic expression where the expression is composed of three terms and represents the area of a square when graphed. These terms include a square of the first term, double the product of the first and second terms, and a square of the second term. The ability to recognize and work with these expressions is crucial in algebra, as it strengthens the understanding of how geometric interpretations can be applied to algebraic problems.

Factoring Algebraic Expressions

The process of factoring algebraic expressions involves breaking down complex expressions into products of simpler factors. When factoring a perfect square trinomial, you are essentially reversing the process of squaring a binomial.

Let's take the perfect square trinomial \(a^2 + 2ab + b^2\). To factor it, we identify the squares \(a^2\) and \(b^2\), and if the middle term is twice the product of a and b, we can conclude that the trinomial factors into \(a+b\)^2. Conversely, if we start with a trinomial like \(x^2 - 4x + 4\), where the first and last terms are squares and the middle term is twice the product of the square roots of the first and last terms (in this case, 2 * 2 * x), it factors into \(x-2\)^2.

Understanding how to factor such expressions is vital, as it greatly simplifies solving quadratic equations and helps make the connections between algebraic solutions and their geometric interpretations.