Problem 28 - Elementary and Intermediate Algebra

Question

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 16-40 z+25 z^{2} $$

Step-by-Step Solution

Answer

The expression factors to \( (5z - 4)^2 \).

Step 1: Arrange Terms

Reorder the terms in the expression in descending powers of \( z \). The expression becomes \( 25z^2 - 40z + 16 \).

Step 2: Identify a Possible Method

Recognize that this is a quadratic trinomial in the form of \( ax^2 + bx + c \), with \( a = 25, b = -40, \text{ and } c = 16 \). Since the leading coefficient and constant term are perfect squares, try to factor by recognizing it as a perfect square trinomial.

Step 3: Check for a Perfect Square Trinomial

For a trinomial \( a^2 - 2ab + b^2 = (a-b)^2 \), test if \( 25z^2 - 40z + 16 \) fits this form. Here, \( a = 5z \) and \( b = 4 \), so: \[ (5z)^2 = 25z^2 \] \[-2(5z)(4) = -40z \] \[ 4^2 = 16 \] Since all conditions hold, it is a perfect square trinomial.

Step 4: Write the Factored Form

Since \( 25z^2 - 40z + 16 \) is a perfect square trinomial, it factors to \( (5z - 4)^2 \).

Key Concepts

Perfect Square Trinomial

Perfect square trinomials are specific kinds of quadratic expressions. They come from squaring a binomial. The standard form of a perfect square trinomial is \( a^2 - 2ab + b^2 = (a-b)^2 \). When you see an expression like this, you can simplify your work by directly recognizing and utilizing this form to factor quickly.
To identify a perfect square trinomial, check for three conditions:

• The first term should be a perfect square, represented by \( a^2 \).
• The last term should also be a perfect square, usually shown as \( b^2 \).
• The middle term must be equal to twice the product of \( a \) and \( b \), or \(-2ab\) if it's a perfect square trinomial in the subtraction form.

Each part must match these guidelines. If the expression fits, it’s a perfect square trinomial that can be factored into \((a-b)^2\). This saves a lot of time compared to other factoring methods.

Quadratic Trinomials

Quadratic trinomials are expressions in the form \( ax^2 + bx + c \). Recognizing these forms can help steer factoring strategies. Quadratic trinomials can be simplified by recognizing certain patterns in their terms.
Every quadratic trinomial can potentially be factored into two binomials. The signs and specific terms involved determine the most suitable factoring technique.

• Look for key indicators like perfect squares and structured numeric relationships between coefficients.
• Perfect square trinomials and difference of squares are key patterns.

The process becomes smoother if you can initially rearrange or rewrite the expression so that it's easier to recognize these connections between terms. Always start by examining if the quadratic trinomial fits neatly into an identifiable form such as a perfect square trinomial for simplified factoring.

Factoring Methods

Factoring is the process of breaking down an expression into simpler components that multiply to the original expression. Different methods apply to different types of expressions. Here are the main techniques:

• **Common Factor Method:** This is the simplest and involves factoring out the greatest common factor from all terms. It's the most basic step that should be attempted.
• **Factoring by Grouping:** This is useful when you have four terms. You regroup pairs of terms and factor them individually.
• **Special Factoring Formulas:** Use these for specialized expressions such as perfect square trinomials or the difference of squares. Recognizing these forms can shortcut the factoring process.
• **Quadratic Formula:** In some cases where simple techniques do not fit, use the quadratic formula. However, this is more calculation-intensive.

Factoring is like solving a puzzle. Recognizing the type of expression you're working with can save time and effort, allowing you to apply the right methods efficiently.