Factoring Perfect Square Trinomials – College Algebra Study Guide

Q1. Factor each of the following perfect square trinomials:

• $x^2 + 10x + 25$

• $x^2 - 6x + 9$

• $9x^2 + 12x + 4$

Background

Topic: Factoring Perfect Square Trinomials

This question tests your ability to recognize and factor trinomials that are the result of squaring a binomial. These trinomials have a special structure and can be factored using patterns.

Key Terms and Formulas

• Perfect Square Trinomial: A trinomial of the form $a^2 + 2ab + b^2$ or $a^2 - 2ab + b^2$.

• Factoring Pattern:

$a^2 + 2ab + b^2 = (a + b)^2$

$a^2 - 2ab + b^2 = (a - b)^2$

Step-by-Step Guidance

1. Identify the first and last terms of each trinomial. Check if they are perfect squares (e.g., $x^2$ and $25$in$x^2 + 10x + 25$).

2. Find the square roots of the first and last terms. For $x^2$, the square root is $x$; for $25$, it is $5$.

3. Check if the middle term matches $2ab$ or $-2ab$, where $a$ and $b$ are the square roots you found. For $x^2 + 10x + 25$, $2ab = 2 \times x \times 5 = 10x$.

4. If the pattern matches, write the trinomial as the square of a binomial: $(a + b)^2$ or $(a - b)^2$.

Try solving on your own before revealing the answer!

Q2. Is it a perfect square trinomial?

• Are the first and last terms perfect squares? If so, list the square roots.

• Is the middle term twice the product of the square roots found in step 1?

• If the answer to both is yes, the trinomial is a perfect square trinomial.

Background

Topic: Identifying Perfect Square Trinomials

This question helps you practice recognizing the structure of perfect square trinomials before factoring.

Key Terms and Formulas

• Perfect Square: A number or expression that is the square of another number or expression.

• Middle Term: Should be $2ab$ or $-2ab$ for a perfect square trinomial.

Step-by-Step Guidance

1. Look at the first and last terms of the trinomial. Determine if each is a perfect square.

2. Find the square roots of these terms.

3. Multiply the square roots together and double the result. Check if this matches the middle term (with the correct sign).

Try solving on your own before revealing the answer!

Q3. Rebekah is making square potholders for family Christmas presents. The area of the potholder is $9x^2 - 6x + 1$ square inches. Write an expression to represent the side length of the potholder.

Background

Topic: Factoring to Find Side Length

This question asks you to factor a perfect square trinomial to find the expression for the side length of a square, given its area.

Key Terms and Formulas

• Area of a Square: $A = s^2$, where $s$ is the side length.

• Perfect Square Trinomial: $a^2 - 2ab + b^2 = (a - b)^2$

Step-by-Step Guidance

1. Identify the first and last terms: $9x^2$ and $1$. Find their square roots: $3x$ and $1$.

2. Check the middle term: $-6x$. Is it $-2ab$? Calculate $-2 \times 3x \times 1 = -6x$.

3. Since the pattern matches, write the area as $(3x - 1)^2$.

4. The side length is the binomial inside the square: $3x - 1$.

Try solving on your own before revealing the answer!