Special Products Perfect Square Trinomial Solve Notice that if you take ½ of the middle number and square it you get the last number 6 divided by 2 is 3 and 3 2 is 9 When this happens you have a special product ID: 477594
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Slide1
Factoring Perfect Square Trinomials and the Difference of SquaresSlide2
Special Products
Perfect Square Trinomial
Solve
Notice that if you take ½ of the middle number and square it, you get the last number. 6 divided by 2 is 3, and 32 is 9. When this happens you have a special product. This problem factors intoSlide3
(
a + b)² = (a + b)(a + b) = a²
+
ab + ab
+ b² = a² +2ab
+b²(a - b)² = (a - b)(a - b) = a²
-
ab
-
ab + b² = a² -2ab +b²
a²
Factoring Perfect Square Trinomials
a = 3x b = 5
To factor this trinomial , the grouping method can be used. However, if we recognize that the trinomial is a perfect square trinomial, we can use one of the following patterns to reach a quick solution
2ab
b²
For example: (3x + 5)² = (3x)² + 2(3x)(5) + (5)² = 9x² + 30x +25Slide4
Checking for a Perfect Square Trinomial
t² + 10t +25
Check if the first and third terms are both perfect squares with positive coefficients.
If this is the case, identify a and b, and determine if the middle term equals 2ab
The first term is a perfect square: t² = (t)²
The third term is a perfect square: 25 = (5)² The middle term is twice the product of t and 5: 2(t)(5)
t² + 10t +25 Perfect square trinomialSlide5
Checking for a Perfect Square Trinomial
t² + 4t +1
Check if the first and third terms are both perfect squares with positive coefficients.
If this is the case, identify a and b, and determine if the middle term equals 2ab
The first term is a perfect square: t² = (t)²
The third term is a perfect square: 1 = (1)² The middle term is not twice the product of t and 1: 2(t)(1)
t² +
4t + 1
Is not perfect square trinomialSlide6
Remember: A perfect square trinomial is one that can be factored into two factors that match each other (and hence can be written as the factor squared).
This is a perfect square trinomial because it factors into two factors that are the same and the middle term is twice the product of x and 6. It can be written as the factor squared.
Notice that the first and last terms are perfect squares
.
The middle term comes from the outers and inners when
FOILing
. Since they match, it ends up double the product of the first and last term of the factor.
Double the product of
x
and 6Slide7
25y² - 20y + 4
The GCF is 1.
The first and third terms are positive
The first
term
is a perfect square: 25y² = (5y)²
The third term is a perfect square: 4 = (2)²
The middle term is twice the product of 5y and 2: 20y = 2(5y)(2)
Factor as
(5y
-
2)²Slide8
Factored Form of a difference of Squares.
a² - b² = (a – b)(a + b)
y² - 25
The binomial is a difference of squares.= (y)² - (5)²
Write in the form: a² - b², where a = y, b = 5.
= (y + 5)(y – 5)
Factor as (a + b)(a – b)Slide9
When you see two terms, look for the difference of squares. Is the first term something squared? Is the second term something squared but with a minus sign (the difference)?
The difference of squares factors into conjugate pairs!
difference
rhyme for the day
A conjugate pair is a set of factors that look the same but one has a + and one has a – between the terms.
{Slide10
Look for something in common (there is a 5)
Two terms left----is it the difference of squares?
Yes---so factor into conjugate pairs.
Factor Completely:Slide11
Look for something in common
There is a 2p in each term
Three terms left---try trinomial factoring
"unFOILing"
Check by FOILing and then distributing 2p throughSlide12
Factoring the Sum and Difference of Cubes and General Factoring Summary
Factoring a Sum and Difference of Cubes
Sum of Cubes: a³+ b³ = (a + b)(a²
-ab +b²)
Difference of Cubes: a³ - b³ = (a - b)(a² +ab
+b²)Slide13
= (
x +
2)
( (x)² -
(x)(2) + (2)²)
x³ + 8 = (x)³ + (2) ³
x³ and 8 are perfect cubes
-(x)(2)
Square the first term of the binomial
Product of terms in the binomial
2
x
(x + 2)
The factored form is the product of a binomial and a trinomial.
The first and third terms in the trinomial are the squares of the terms within the binomial factor.
Without regard to signs, the middle term in the trinomial is the product of terms in the binomial factor.
Square the last term of the binomial.Slide14
If it's not the difference of squares, see if it is sum or difference of cubes. Is the first term something cubed (to the third power)? Is the second term something cubed?
can be sum or difference here
You must just memorize the steps to factor cubes. You should try multiplying them out again to assure yourself that it works.
The first factor comes from what was cubed
.
square the first term
multiply together but change sign
square the last termSlide15
What cubed gives the first term?
What cubed gives the second term?
Let's try one more:
Try to memorize the steps to get the second factor:
First term squared---multiply together & change sign---last term squared
The first factor comes from what was cubed
.
square the first term
multiply together but change sign
square the last term