Topic 6 Lesson 7
Factoring Special Cases
I CAN…
factor special trinomials.
VOCABULARY
• perfect-square trinomial
MA.912.AR.1.7–Rewrite a polynomial expression as a product of polynomials
over the real number system. Also AR.1.1
MA.K12.MTR.4.1, MTR.5.1, MTR.6.1
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Topic 6 Lesson 7
CRITIQUE & EXPLAIN
Seth and Bailey are given the polynomial 8x2 + 48x + 72 to factor.
A. Analyze each factored expression to see if both are equivalent to the given polynomial.
B. How can the product of different pairs of expressions be equivalent?
C. Represent and Connect Find two other pairs of binomials that are different, but whose
products are equal.
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Topic 6 Lesson 7
? ESSENTIAL QUESTION
What special patterns are helpful when factoring a perfect-square trinomial and the
difference of two squares?
CONCEPTUAL UNDERSTANDING
EXAMPLE 1 Understand Factoring a Perfect Square
What is the factored form of a perfect-square trinomial?
A perfect-square trinomial results when a binomial is squared.
A. What is the factored form of x2 + 14x + 49?
Write the last term as a perfect square.
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Topic 6 Lesson 7
CONCEPTUAL UNDERSTANDING
EXAMPLE 1 Understand Factoring a Perfect Square
What is the factored form of a perfect-square trinomial?
A perfect-square trinomial results when a binomial is squared.
B. What is the factored form of 9x2 – 30x + 25?
Write the first and last terms as a perfect square.
The factored form of a perfect-square trinomial is (a + b)2 when the trinomial
fits the pattern a2 + 2ab + b2, and (a − b)2 when the trinomial fits the pattern
a2 − 2ab + b2.
COMMON ERROR
Be careful to identify the correct values for a and b when factoring special cases. The value of a can be
different from x.
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Topic 6 Lesson 7
EXAMPLE 1 Understand Factoring a Trinomial
Try It!
1. Factor each trinomial.
a. 4x2 + 12x + 9
b. x2 – 8x + 16
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Topic 6 Lesson 7
APPLICATION
EXAMPLE 2 Factor to Find a Dimension
Sasha has a tech store and needs cylindrical
containers to package her voice-activated
speakers. A packaging company makes two
different cylindrical containers. Both are 3 in. high.
The volume information is given for each type of
container. Determine the radius of each cylinder.
How much greater is the radius of one container
than the other?
Formulate
The formula for the volume of a cylinder is V = pr2h , where r is the radius and h is
the height of the cylinder. The height of both containers is 3 in., so both expressions
will have 3p in common.
Factor the expressions to identify the radius of each cylinder.
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Topic 6 Lesson 7
APPLICATION
EXAMPLE 2 Factor to Find a Dimension
Sasha has a tech store and needs cylindrical
containers to package her voice-activated
speakers. A packaging company makes two
different cylindrical containers. Both are 3 in. high.
The volume information is given for each type of
container. Determine the radius of each cylinder.
How much greater is the radius of one container
than the other?
Compute
The expression x2 = x • x, so the radius of the first cylinder is x in.
Factor the expression x2 + 10x + 25 to find the radius of the second cylinder.
The radius of the second cylinder is (x + 5) in.
Find the difference between the radii.
(x + 5) – x = 5
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Topic 6 Lesson 7
APPLICATION
EXAMPLE 2 Factor to Find a Dimension
Sasha has a tech store and needs cylindrical
containers to package her voice-activated
speakers. A packaging company makes two
different cylindrical containers. Both are 3 in. high.
The volume information is given for each type of
container. Determine the radius of each cylinder.
How much greater is the radius of one container
than the other?
Interpret
The larger cylinder has a radius that is 5 in. greater than the smaller one.
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Topic 6 Lesson 7
EXAMPLE 2 Factor to Find a Dimension
Try It!
2. What is the radius of a cylinder that has a height of 3 in. and a volume of
p(27x2 + 18x + 3) in.3?
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Topic 6 Lesson 7
EXAMPLE 3 Factor a Difference of Two Squares
How can you factor the difference of squares using a pattern?
Recall that a binomial in the form a2 – b2 is called the difference of two squares.
A. What is the factored form of x2 – 9?
Write the last term as a perfect
square.
B. What is the factored form of 4x2 – 81?
Write the first and last terms
as perfect squares.
The difference of two squares is a factoring pattern when one perfect square
is subtracted from another. If a binomial follows that pattern, you can factor it
as a sum and difference.
CHECK FOR REASONABLENESS
Determine whether the factoring rule for a difference of two squares makes sense by working backward.
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Topic 6 Lesson 7
EXAMPLE 3 Factor a Difference of Two Squares
Try It!
3. Factor each expression.
a. x2 − 64
b. 9x2 − 100
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Topic 6 Lesson 7
EXAMPLE 4 Factor Out a Common Factor
What is the factored form of 3x3y – 12xy3?
Factor out a greatest common factor of the terms if there is one. Then factor as the
difference of squares.
The factored form of 3x3y − 12xy3 is 3xy(x + 2y)(x − 2y).
HAVE A GROWTH MINDSET
How can you take on challenges with positivity?
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Topic 6 Lesson 7
EXAMPLE 4 Factor Out a Common Factor
Try It!
4. Factor each expression completely.
a. 4x3 + 24x2 + 36x
b. 50x2 – 32y2
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Topic 6 Lesson 7
CONCEPT SUMMARY
Factoring Special Cases of Polynomials
Factoring a Perfect-Square Trinomial
Factoring a Difference of Two Squares
ALGEBRA
WORDS
Use this pattern when the first and last terms
are perfect squares and the middle term is twice
the product of the expressions being squared.
Use this pattern when a binomial can be written
as a difference of two squares. Both terms must
be perfect squares.
NUMBERS
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Topic 6 Lesson 7
Do You UNDERSTAND?
1.
? ESSENTIAL QUESTION What special patterns are helpful when factoring a
perfect-square trinomial and the difference of two squares?
2. Error Analysis A student says that to factor x2 – 4x + 4, you should use the pattern of
a difference of two squares. Explain the error in the student’s thinking.
3. Vocabulary How is a perfect square trinomial similar to a perfect square number? Is it
possible to have a perfect square binomial? Explain
4. Use Patterns and Structure How is the pattern for factoring a perfect-square trinomial
like the pattern for factoring the difference of two squares? How is it different?
5. Communicate and Justify Why is it important to look for a common factor before
factoring a trinomial?
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Topic 6 Lesson 7
Do You KNOW HOW?
Identify the pattern you can use to factor each expression.
6. 4x2 – 9
7. x2 + 6x + 9
8. 9x2 – 12x + 4
9. 5x2 – 30x + 45
10. 100 – 16y2
11. 3x2 + 30x + 75
Write the factored form of each expression.
12. 49x2 − 25
13. 36x2 + 48x + 16
14. 3x3 – 12x2 + 12x
15. 72x2 – 32
16. What is the side length of the square shown below?
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