Problem 73 - Elementary and Intermediate Algebra

Question

Factor. $$ 9 y^{2}-24 y+16 $$

Step-by-Step Solution

Answer

The expression factors to \((3y - 4)^2\).

Step 1: Identify Common Factors

The expression is \(9y^2 - 24y + 16\). First, check if there's any common factor across the terms 9, -24, and 16. Since there isn't a common factor other than 1, we proceed to the next step.

Step 2: Check Perfect Square Trinomial

This expression resembles a perfect square trinomial. Recall that a perfect square trinomial follows the formula \(a^2 - 2ab + b^2 = (a-b)^2\). We will check if it fits in this form.

Step 3: Take Square Roots of First and Last Terms

Compute the square roots of the first and last terms: \(\sqrt{9y^2} = 3y\) and \(\sqrt{16} = 4\). These indicate the potential values of \(a\) and \(b\) in the perfect square formula.

Step 4: Check Middle Term Using 2ab

Calculate \(-2 \, \times \, 3y \, \times \, 4 = -24y\). This matches the middle term of the expression. This confirms that the trinomial is a perfect square.

Step 5: Write in Factored Form

Since the conditions for a perfect square are met, the expression \(9y^2 - 24y + 16\) can be factored as \((3y - 4)^2\).

Key Concepts

Perfect Square Trinomials

A perfect square trinomial is a special type of polynomial that takes the form \(a^2 - 2ab + b^2\), which simplifies to \((a-b)^2\). These trinomials are called "perfect square" because they can be expressed as the square of a single binomial.

Identifying a perfect square trinomial involves recognizing the specific pattern:

• The square of the first term, \(a^2\)
• A middle term, \(-2ab\), which is twice the product of the two distinct terms
• The square of the last term, \(b^2\)

In the example given, \(9y^2 - 24y + 16\), we checked if this could fit the formula. We found \(a = 3y\) and \(b = 4\), which confirmed its identity as a perfect square.

Recognizing these trinomials helps simplify expressions and solve equations more easily, making it an essential tool in algebra.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. They are foundational in algebra and can be as simple as \(x + 2\) or as complex as polynomial expressions like \(9y^2 - 24y + 16\).

In general, algebraic expressions consist of:

• Terms: Parts of the expression separated by addition or subtraction, each consisting of coefficients and variables
• Coefficients: The numerical part of the terms, like 9, 24, or 16 in the given expression
• Variables: Symbols that represent unknown values, such as \(y\)

Understanding these components enables manipulation and simplification of expressions through various algebraic techniques. In our example, recognizing the lack of common factors was key to identifying the structure of the expression as a perfect square trinomial.

Mastery of algebraic expressions allows students to tackle more advanced topics, including quadratic equations and polynomial functions.

Factoring Techniques

Factoring is a process of rewriting an expression as a product of its factors. It's one of the core techniques used to simplify expressions and solve equations. For the trinomial \(9y^2 - 24y + 16\), we used the perfect square trinomial factoring technique.

Different techniques exist depending on the type of polynomial:

• Common factoring, which involves taking out the greatest common factor (GCF) of all terms
• Factoring by grouping, useful for polynomials with four or more terms
• Recognizing special product forms, like the difference of squares or perfect square trinomials

To factor \(9y^2 - 24y + 16\), recognizing it as a perfect square trinomial allowed us to write it as \((3y - 4)^2\). This method relies on identifying patterns rather than complex computations. Once mastered, factoring can greatly simplify solving equations, especially quadratic ones, by revealing roots or solutions easily.