6.4 Perfect-Square Trinomials and Differences of Squares 2 = ( +

6.4 Perfect-Square Trinomials and Differences of Squares
Perfect-Square Trinomials
Differences of Squares
More Factoring by Grouping
Solving Equations
Recall using FOIL
(𝑥 + 3)! = (𝑥 + 3)(𝑥 + 3)
(𝐴 + 𝐵)! =
(𝐴 − 𝐵)! =
(𝑥 + 7)!
(2𝑥 + 5)!
(𝑥 − 3)!
(2𝑥 − 5)!
Perfect Squares:
1-10??????
Perfect-Square Trinomials
𝑥 ! + 6𝑥 + 9
To recognize a Perfect-Square Trinomial
1. The first and last terms must be of the form
𝐴! 𝑎𝑛𝑑 𝐵 ! ,
2. Neither 𝐴! 𝑛𝑜𝑟 𝐵 ! is being subtracted,
3. The middle term must be either 2𝐴𝐵 𝑜𝑟 − 2𝐴𝐵.
Ex. 1 Factor:
(ALWAYS look for a common term first!)
𝑥 ! + 10𝑥 + 25
4𝑥 + 16 + 3𝑥 !
100𝑦 ! + 81 − 180𝑦
2
Ex. 2 Factor:
𝑥 ! − 10𝑥 + 25
16𝑦 ! + 49 + 56𝑦
−20𝑥𝑦 + 4𝑦 ! + 25𝑥 !
Ex. 3 Factor: −4𝑦 ! − 144𝑦 ! + 48𝑦 !
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Differences of Squares
Recall from FOIL
Ex. 4 Factor:
(𝑥 + 3)(𝑥 − 3)
9𝑡 ! − 64
(2𝑥 − 1)(2𝑥 + 1)
25 − 𝑥 !
−4𝑥 !" + 36
Factoring a Difference of Two Squares
Note:
𝐴! − 𝐵! = (𝐴 + 𝐵)(𝐴 − 𝐵)
𝐴! + 𝐵! =? ? ? ? ?
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Ex. 5 Factor:
𝑥! − 9
25𝑦 ! − 49𝑥 !
Factoring is complete when NO factor can be factored further
Ex. 6 Factor:
5 − 5𝑥 ! 𝑦 !
16𝑥 ! 𝑦 − 81𝑦
More Factoring By Grouping
Ex. 7
Factor:
𝑥 ! + 3𝑥 ! − 4𝑥 − 12
5
(In four or more terms there may be a perfect square.)
Ex. 8 Factor:
𝑥 ! + 6𝑥 + 9 − 𝑦 !
𝑎! − 𝑏 ! + 8𝑏 − 16
Solving Equations
Ex. 9 Solve:
𝑥 ! + 3𝑥 ! = 4𝑥 + 12
(see #7)
Ex. 10 Find the zeros of the function given by :
𝑓 𝑥 = 𝑥! + 𝑥! − 𝑥 − 1
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Use the graphing calculator to look at the roots/zeros for the last two
equations/functions
𝑥 ! + 3𝑥 ! = 4𝑥 + 12
𝑓 𝑥 = 𝑥! + 𝑥! − 𝑥 − 1
Some equations/functions cannot be factored and are said to be
prime….however we can look at their graph and get an approximate
root/zero.
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