Factoring Special Cases
Key Concepts
- Understand factoring a perfect square
- Factors find a dimension
- Factor a difference of two squares
Understand factoring a Perfect Square Trinomial
Concept
What is the factored form of a perfect square trinomial?
A perfect square trinomial results when a binomial is squared.
Examples
A. What is the factored form of x2 + 14x + 49?
Write the last term as a perfect square.
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 trinomials fit the pattern a2 – 2ab + b2.
Application: 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 the cylinder is V= pi*(r2) *h, where r is the radius and h is the height of the cylinder. The height of both the containers is 3 in., so both expressions will have 3*pi in common.
Factor the expressions to identify the radius of each cylinder.
Compute:
The expression x2 = x*x, so the radius of the first cylinder is x in.
Factor the expression x2 + 10x + 5 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
Interpret: The larger cylinder has a radius that is 5 in. greater than the smaller one.
Factor a Difference of Two Squares
Concept
How can you factor the difference of two squares using a pattern?
Recall that a binomial in the form a2 – b2 is called the difference of two squares.
(a – b) (a + b) = a2 – ab + ab – b2 = a2 – b2
Examples
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 a perfect square.
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.
Factor out a Common Factor
What is the factored form of 3(x3) y – 12xy3?
Factor out the greatest common factor of the terms if there is one. Then factor as the difference of squares.
The factored form of 3(x3) y – 12xy3 is 3xy (x + 2y) (x – 2y).
Questions
Question 1
Write the factored form of each trinomial.
1. 4x2 + 12x + 9
(2x)2 + 2(2x) (3) + 32
Using the pattern (a + b)2 = a+2 + 2ab + b2,
(2x)2 + 2(2x) (3) + 32 = (2x + 3)2
2. x2 – 8x + 16
x2 – 2(x)(4) + 42
Using the pattern (a – b)2 = a2 – 2ab + b2,
x2 – 2(x)(4) + 4+2 = (x – 4)2
3. x2 – 64
x2 – 82
Using the pattern (a – b) (a + b) = a2 – b2,
x2 – 82 = (x – 8) (x + 8)
4. 9x2 – 100
(3x)2 – 102
Using the pattern (a – b) (a + b) = a2 – b2,
(3x)2 – 102 = (3x – 10) (3x + 10)
5. 4x3 + 24x2 + 36x
Take out x,
x (4x2 + 24x + 36)
= x((2x)2 + 2(2x) (6) + 62)
= x (2x + 6)2 = x (2x + 6) (2x + 6)
6. 50x2 – 32y2
Take out 2,
2(25x2 – 16y2)
= 2 ((5x)2 – (4y)2)
= 2(5x – 4y) (5x + 4y)
Question 2
What is the radius of a cylinder that has a height of 3 in. and a volume of pi (27x2 + 18x + 327x2 + 18x + 3) cubic inches?
Solution:
Volume of a cylinder: V = pi*(r2) *h
pi*(r2) *h = pi (27x2 + 18x + 3) (h = 3in.)
(r2) = 9x2 + 6x + 1
Let’s factor 9x2 + 6x + 1 to find the value of r.
(3x)2 + 2(3x) (1) + 12
= (3x + 1)2
So, r2 = (3x + 1)2 which means r = (3x + 1) in.
Question 3
The area of a rectangular rug is 49x2 – 25y2 square in. Use factoring to find the possible dimensions of the rug. How are the side lengths related? What value would you need to subtract from the longer side and add to the shorter side for the rug to be a square?
Solution:
Area = 49x2 – 25y2 = (7x)2 – (5y)2 = (7x + 5y) (7x – 5y)
Length of longer side (l) = 7x + 5y
Length of shorter side (s) = 7x – 5y
l – s = (7x + 5y) – (7x – 5y) = 10y
Relation between longer side (l) and shorter side(s): l = s + 10y
7x + 5y – 5y = 7x
7x – 5y + 5y = 7x
5y would need to be subtracted from the longer side and 5y would need to be added to the shorter side for rug to be a square.
Key Concepts Covered
Exercise
Factor:
- x2 + 16x + 64
- x2 – 16x + 64
- x2 – 36
- 4x3 + 24x2 + 36x
- 9x2 – 100
- 100 – 16y2
- 5x2 – 30x + 45
- 49x2 – 25
- 16x4 – y4
- x2 – 1/9
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