What’s a Perfect Square?—Text Version Perfect Squares … What Are They? A perfect square is an integer that can be made by multiplying two of the same integers together. For example, 36 is a perfect square. 36 can be factored into 6 times 6. Since the factored terms are the same, 36 is a perfect square. Is 33 a perfect square? 33 can be factored into 3 times 11, but it cannot be factored into two of the same integer. Is 4 a perfect square? Yes, 4 can be factored into 2 times 2. But what about variables? Can they be perfect squares too? Yes! x to the 10th power can be factored into x to the 5th power times x to the 5th power; so x to the 10th is a perfect square. What about x to the 9th power? x to the 9th power can be factored into x to the 4th power times x to the 5th power, but it cannot be factored into two of the same expressions with integer exponents.

ALGEBRA I : MODULE 08 : FACTORING : 08.06 DIFFERENCE OF PERFECT SQUARES

Question 1 Question 2 Question 3

Factoring a − b

Practice identifying whether an expression is or is not the difference of two perfect squares!

4x − 9y1. Are there only two terms?

Check Your Answer

Yes

2. Are both terms perfect squares?

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First, select the chalkboard to review what you alreadyknow.

Chalkboard—Text Version

A difference of squares binomial is the product of two binomials that have the same first term andopposite second terms.

(a + b)(a − b) = a − b

The first term is a perfect square.The second term is a perfect square.The terms are separated by subtraction.

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Check Your Answer

Yes

3. Are the terms separated by a subtraction sign?

Check Your Answer

Yes. Because the answer is yes to all of the questions, this is a difference of perfect squares.

Now that you are able to easily identify whether an expression is the difference of two squares, it’s time tospice things up!

Factoring the Difference of Two SquaresOnce you know an expression is the difference of two squares, how can you factor it?

Take a look at the table below.

Difference of Two Squares Factored Form

x − 4 (x + 2)(x − 2)

x − 36 (x + 6)(x − 6)

x − 9 (x + 3)(x − 3)

Let's make some interesting and important observations based on the table above. Answer each question,then select the question to see if you are correct.

What do you notice about the signs of the factored binomials?

One is positive, and one is negative.

What do you notice about the first term of each factored binomial?

It is the same term and the square root of the first term of the original binomial.

What do you notice about the second term of each factored binomial?

It is the same term and the square root of the second term of the original binomial.

a − b = (a + b)(a − b)

So, to factor a difference of two squares binomial:

find the square root of the first term—this will be the first term of each binomial factor.find the square root of the second term—this will be the second term of each binomial factor.one binomial will have an addition sign, and one will have a subtraction sign.

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Example 1

Take a look at the example below.

Factor completely: 9x − 4

Check out a couple more examples of factoring the difference of two squares before trying some onyour own!

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Factoring 9x − 4—Text Version

Factor completely: 9x − 4.

Step 1 is to factor out the GCF. This binomial does not have a GCF, so you may move toStep 2.

Step 2: Determine whether this is a Difference of Two Squares. 9 times x squared is aperfect square with terms of 3 times x. 4 is a perfect square with terms of 2. The binomial isseparated by subtraction, so it is the difference of squares.

Step 3: Factor completely. Display two pairs of parenthesis. Write the 3 times x as the leftterm within each pair of parenthesis. Write the 2 as the right term in each pair ofparenthesis. Within the first set of parenthesis, add an addition operator between the 3times x term and the 2. Within the second set of parenthesis, add a subtraction operatorbetween the 3 times x terms and the 2. This provides the expression the quantity 3 times xplus 2 end quantity times the quantity 3 times x minus 2 end quantity.

Check the answer by simplifying the expression. This provides 9 times x squared minus 4,so the factorization is correct.

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Select the chalkboard to practice factoring some more binomials.

Factoring Out the GCF Sometimes, when factoring the difference of two squares, you have to factor out a GCF before you see two perfect square terms.

Example 2

Example

Factor completely: x + 4

Step 1: Factor out the GCF

This binomial does not have a GCF, so you may move to step 2.

Step 2: Determine whether this is a Difference of Two Squares

Is this expression the difference of two squares? Select each part of the expressionbelow to reveal the answer.

x + 4 is the sum of two perfect squares.

Can the sum of two squares be factored?

Do any of these factors have a product of x + 4?

Let's try it out

(x + 2)(x − 2) (x + 2)(x + 2) (x − 2)(x − 2)

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Expression—Text Version

The terms and operator in the expression x squared plus 4 are selectable. If the xsquared is selected, the text "x times x" and "the first term is a perfect square" isdisplayed with an arrow pointing to the x squared term. If the addition sign isselected, the text "addition sign" is displayed with a blue arrow pointing to theaddition sign. If the 4 is selected, the text "2 times 2" and "the second term is aperfect square" is displayed with a green arrow pointing to the 4 term.

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(x + 2)(x − 2) (x + 2)(x + 2) (x − 2)(x − 2)

x − 2x + 2x − 4 x − 4 No

x + 2x + 2x + 4x + 4x + 4 No

x − 2x − 2x + 4 x − 4x + 4No

As you can see, none of these products equals x + 4, and therefore, the sum oftwo squares (without a GCF) is always prime.

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Example 3

Example

Factoring a Tricky Binomial

Example Factor completely: 5x − 25

Step 1: Factor out the GCF

This binomial does have a GCF. You can factor out a 5 from both terms.

5x − 25 = 5(x − 5)

Now, factor the remaining binomial in parentheses.

Step 2: Determine whether this is a Difference of Two Squares

Is the expression inside the parentheses the difference of two squares? Select each partof the expression below to reveal the answer.

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Expression—Text Version

The terms and operator in the expression 5 times the quantity x squared minus 5are selectable. If the x squared is selected, the text "x times x" and "the first termis a perfect square" is displayed with an arrow pointing to the x squared term. Ifthe subtraction sign is selected, the text "subtraction sign" is displayed with ablue arrow pointing to the subtraction sign. If the 5 is selected, the text "thesecond term is not a perfect square" is displayed with a green arrow pointing tothe 5 term.

The expression within the parentheses is not the difference of two squaresbecause the second term is not a perfect square.

At this point, the expression cannot be factored further. There are times when theonly thing you can do is factor out a GCF. The factored form of 5x − 25 is 5(x −5).

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ALGEBRA I : MODULE 08 : FACTORING : 08.06 DIFFERENCE OF PERFECT SQUARES

Example 1 Example 2 Example 3

Factoring Numerical ExpressionsNow that you can recognize if an expression is a difference of squares, you can use that information alongwith mental math to help you solve other mathematical problems. Let’s see just how easy it is.

What do you notice about this expression: 28 − 22 ? At first thought, you might rush to your calculator, butactually it can be evaluated with ease. Recognizing the structure of an expression can help you rewrite it.

Did you quickly recognize this as being the difference of two squares?

a − b = (a + b)(a − b)

Using that information, you can rewrite the initial problem so that you can solve using mental math.

28 − 22 = (28 + 22)(28 − 22) = (50)(6) = 300

See how easy that was? No calculator required. Let’s look at two more examples.

Factoring Polynomials with Multiple MethodsPreviously, you have learned to factor out a GCF and to factor using the grouping method. You alwayschecked to make sure you had factored completely. Sometimes after using one factoring method, anothermethod can arise. You can combine factoring the GCF with grouping and, in some cases, with difference ofperfect squares. Let’s look at a few examples.

Factor x + 2x − 9x − 18

1. First, look for a GCF. There is not one in this example.2. Since there are four terms, you can factor using the grouping method.

x + 2x − 9x − 18

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53 − 47

= (53 + 47)(53 − 47)= (100)(6)= 600

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15 − 10

= (15 + 10)(15 − 10)= (25)(5)= 125

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x (x + 2) − 9(x + 2)

(x − 9)(x + 2)3. Check to make sure each binomial is factored completely. The first binomial has the correct

structure to continue factoring using the difference of two squares method.

(x − 9)(x + 2)

(x + 3)(x − 3)(x + 2)

Wow! When this polynomial is factored, you end up with three binomial factors.

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Select the chalkboard to practice factoring some more expressions.

Activity 1—Text VersionFactor each binomial completely. When you’re done, select Check Your Answers to see how you did.

Question 1: Factor a squared minus 9

a. the quantity a minus three end quantity times the quantity a minus three end quantityb. the quantity a plus three end quantity times the quantity a plus three end quantityc. the quantity a minus three end quantity times the quantity a plus three end quantityd. the quantity a minus nine end quantity times the quantity a plus nine end quantity

Question 2: Factor x squared minus y squared

a. the quantity x minus y end quantity times the quantity x minus y end quantityb. the quantity x minus y end quantity times the quantity x plus y end quantityc. the quantity x minus y squared end quantity times the quantity x plus y squared end quantityd. the quantity x plus y end quantity times the quantity x plus y end quantity

Question 3: Factor eighty one times p squared minus twenty five

a. the quantity nine times p minus five end quantity times the quantity nine times p minus five end quantityb. the quantity nine times p minus five end quantity times the quantity nine times p plus five end quantityc. the quantity nine times p minus twenty five end quantity times the quantity nine times p plus twenty five end

quantityd. the quantity nine times p plus twenty five end quantity times the quantity nine plus twenty five end quantity

Question 4: Factor a squared minus twenty five

a. the quantity a minus five end quantity times the quantity a plus five end quantityb. the quantity a minus five end quantity times the quantity a minus five end quantityc. the quantity a plus twenty five end quantity times the quantity a plus twenty five end quantityd. the quantity a minus twenty five end quantity times the quantity a plus twenty five end quantity

Question 5: Factor t squared plus four

a. the quantity t plus two end quantity times the quantity t plus two end quantityb. the quantity t plus two end quantity times the quantity t minus two end quantityc. the quantity t minus two end quantity time the quantity t minus two end quantityd. the binomial is prime

Question 6: Factor x squared minus sixteen times y squared

a. the quantity x minus four times y end quantity times the quantity x plus four times y end quantityb. the quantity x minus four times y end quantity times the quantity x minus four times y end quantityc. the quantity x plus sixteen times y end quantity times the quantity x plus sixteen times y end quantityd. the quantity x minus sixteen times y end quantity times the quantity x plus sixteen times y end quantity

Question 7: Factor p squared minus forty nine

a. the quantity p minus seven end quantity squaredb. the quantity p plus seven end quantity squaredc. the quantity p squared minus seven end quantity times the quantity p minus seven end quantityd. the quantity p plus seven end quantity times the quantity p minus seven end quantity

Question 8: Factor x to the sixth power minus twelve

a. the quantity x minus two end quantity raised to the sixth powerb. the quantity x cubed plus four end quantity times the quantity x cubed minus three end quantityc. the quantity x cubed plus six end quantity times the quantity x cubed minus six end quantityd. the binomial is prime

Question 9: Factor four times x squared minus nine times y squared

a. the quantity two times x plus three times y end quantity times the quantity two times x plus three times y endquantity

b. the quantity two times x minus three times y end quantity times the quantity two times x minus three times yend quantity

c. the quantity two times x plus three times y end quantity times the quantity two times x minus three times y endquantity

d. The binomial is prime.

Question 10: Factor p squared minus four

a. the quantity p minus two end quantity times the quantity p plus two end quantityb. the quantity p minus two end quantity times the quantity p minus two end quantityc. the quantity p plus four end quantity times the quantity p plus four end quantityd. the quantity p minus four end quantity times the quantity p plus four end quantity

Check Your Answers

Here are the answers!

Question 1: Factor a squared minus 9

Correct Answer: c. the quantity a minus three end quantity times the quantity a plus three end quantity

Question 2: Factor x squared minus y squared

Correct Answer: b. the quantity x minus y end quantity times the quantity x plus y end quantity

Question 3: Factor eighty one times p squared minus twenty five

Correct Answer: b. the quantity nine times p minus five end quantity times the quantity nine times p plusfive end quantity

Question 4: Factor a squared minus twenty five

Correct Answer: a. the quantity a minus five end quantity times the quantity a plus five end quantity

Question 5: Factor t squared plus four

Correct Answer: d. the binomial is prime

Question 6: Factor x squared minus sixteen times y squared

Correct Answer: a. the quantity x minus four times y end quantity times the quantity x plus four times y endquantity

Question 7: Factor p squared minus forty nine

Correct Answer: d. the quantity p plus seven end quantity times the quantity p minus sevenend quantity

Question 8: Factor x to the sixth power minus twelve

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Correct Answer: d. the binomial is prime

Question 9: Factor four times x squared minus nine times y squared

Correct Answer: c. the quantity two times x plus three times y end quantity times the quantity two times xminus three times y end quantity

Question 10: Factor p squared minus four

Correct Answer: a. the quantity p minus two end quantity times the quantity p plus two end quantity

ALGEBRA I : MODULE 08 : FACTORING : 08.06 DIFFERENCE OF PERFECT SQUARES

Unless Otherwise Noted All Content © 2020 Florida Virtual School

Practice

Activity 2Question 1. How can you use difference of squares to evaluate 34 − 26 ?

Show me

34 − 26 = (34 + 26)(34 − 26) = (60)(8) = 480

Question 2. How can you use difference of squares to evaluate 40 − 30 ?

Show me

40 − 30 = (40 + 30)(40 − 30) = (70)(10) = 700

Question 3. Factor completely 4x − 4x − x + 1.

Show me

4x − 4x − x + 1 = 4x (x − 1) − 1(x − 1) = (4x − 1)(x − 1) = (2x − 1)(2x + 1)(x − 1)

Question 4. Factor completely 2x − 162.

Show me

2x − 162 = 2(x − 81) = 2(x + 9)(x − 9)

Question 5. Factor completely 16x − 100y .

Show me

16x − 100y = 4(4x − 25y ) = 4(2x + 5y)(2x − 5y)

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08.06 Review

Follow these steps when factoring the difference of two squares:

Step 1: Factor Out the GCFStep 2: Determine Whether the Binomial is a Difference of Two Squares

The expression must have two terms, both of which are perfect squares, and be separated by asubtraction sign.

Step 3: Factor Completely

Find the square root of each term, and place them in the difference of two squares pattern.

a − b = (a + b)(a − b)

Step 4: Check Your Factors

If you arrive back at the original expression, you know that you factored correctly!

You can use the difference of perfect squares and mental math to help evaluate numerical expressionsthat take this form. You can also combine factoring methods like grouping and difference of perfectsquares when factoring higher degree polynomials.

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