Download Intro: Factoring perfect square trinomials

12/8/2016
(67) Factoring quadratics: Perfect squares | Factoring quadratics: Perfect squares | Polynomial factorization | Algebra I | Khan Academy
Factoring quadratics: Perfect squares
Learn how to factor quadratics that have the "perfect square" form. For example, write x²
+6x+9 as (x+3)².
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Factoring a polynomial involves writing it as a product of two or more
polynomials. It reverses the process of polynomial multiplication.
In this article, we'll learn how to factor perfect square trinomials using special
patterns. This reverses the process of squaring a binomial, so you'll want to
understand that completely before proceeding.
Intro: Factoring perfect square trinomials
To expand any binomial, we can apply one of the following patterns.
2
2
2
(a + b) =a + 2ab +b
2
2
2
(a − b) =a − 2ab +b
[Where do these patterns come from?]
Note that in the patterns, a and b can be any algebraic expression. For
2
example, suppose we want to expand(x + 5) . In this case, a
and so we get:
2
2
= x and b = 5,
2
(x + 5) =x + 2(x)(5) + (5)
2
= x + 10x + 25
2
You can check this pattern by using multiplication to expand (x + 5) . [I'd like to see this expansion, please!]
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The reverse of this expansion process is a form offactoring. If we rewrite the
equations in the reverse order, we will have patterns for factoring polynomials
2
of the forma
2
2
± 2ab + b .
2
2
a + 2ab +b = (a + b)
2
2
2
a − 2ab +b = (a − b)
2
We can apply the first pattern to factor x
have a
+ 10x + 25. Here we
= x and b = 5.
2
2
2
x + 10x + 25 =x + 2(x)(5) + (5)
2
= (x + 5)
Expressions of this form are called perfect square trinomials. The name
reflects the fact that this type of three termed polynomial can be expressed as
a perfect square!
Let's take a look at a few examples in which we factor perfect square
trinomials using this pattern.
2
Example 1: Factoring x
+ 8x + 16
Notice that both the first and last terms are perfect
2
2
squares: x
2
= (x) and 16 = (4) . Additionally, notice that the middle term is
two times the product of the numbers that are squared: 2(x)(4) = 8x.
This tells us that the polynomial is a perfect square trinomial, and so we can
use the following factoring pattern.
2
2
2
a + 2ab +b = (a + b)
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In our case, a
= x and b = 4. We can factor our polynomial as follows:
2
2
2
x + 8x + 16 = (x) + 2(x)(4) + (4)
2
= (x + 4)
2
We can check our work by expanding (x + 4) :
2
2
2
(x + 4) = (x) + 2(x)(4) + (4)
2
= x + 8x + 16
[Can this be factored another way?]
Check your understanding
2
1) Factor x
+ 6x + 9.
(x − 3)(x + 3)
2
(x + 3)(x + 3) or (x + 3)
2
(x − 3)(x − 3) or (x − 3)
Check
[I need help!]
2
2) Factor x
− 6x + 9.
(x − 3)(x + 3)
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2
(x + 3)(x + 3) or (x + 3)
2
(x − 3)(x − 3) or (x − 3)
Check
[I need help!]
2
3) Factor x
+ 14x + 49.
Check
[I need help!]
2
Example 2: Factoring 4x
+ 12x + 9
It is not necessary for the leading coefficient of a perfect square trinomial to
be 1.
2
For example, in 4x
+ 12x + 9, notice that both the first and last terms are
2
2
perfect squares: 4x = (2x) and9 = (3) . Additionally, notice that the middle
2
term is two times the product of the numbers that are
squared: 2(2x)(3)
= 12x.
2
Because it satisfies the above conditions, 4x
+ 12x + 9is also a perfect
square trinomial. We can again apply the following factoring pattern.
2
2
2
a + 2ab +b = (a + b)
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In this case, a
= 2x and b = 3. The polynomial factors as follows:
2
2
2
4x + 12x + 9 = (2x) + 2(2x)(3) + (3)
2
= (2x + 3)
2
We can check our work by expanding (2x + 3) .
[Can this be factored another way?]
Check your understanding
2
4) Factor 9x
+ 30x + 25.
2
(3x − 5)
2
(3x + 5)
(9x + 5)(x + 5)
Check
[I need help!]
2
5) Factor 4x
− 20x + 25.
Check
[I need help!]
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Challenge problems
4
2
6*) Factor x
+ 2x + 1.
Check
[I need help!]
2
7*) Factor 9x
2
+ 24xy + 16y .
Check
[I need help!]
Factoring quadratics: Perfect squares
Factoring perfect squares
Factoring quadratics: Perfect squares
Identifying perfect square form
Factoring perfect squares: common factor
Factoring perfect squares: negative common factor
Factoring perfect squares: missing values
Factoring perfect squares: shared factors
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Practice: Factor special products (basic)
Practice: Factor perfect squares
Factoring quadratics in any form
Next tutorial
Factoring polynomials with special product for...
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