Algebra Test Prep and Review
Unit 5 – Polynomial Functions
Special
Special
Binomial
Binomial
Products
Products
• Special
• Special
binomial
binomial
products
products
– squaring
– squaring
binominals
binominals
Special
Special
Products
Products
Formula
Formula
InitialInitial
Expansion
Expansion
Example
Example
2
2
2
2
2
2 2
2
2 2 (a 2+ b)(a
(a
+
–
b)
b)(a
=
a
–
b)
–
ab
=
a
+
–
ba
ab
–
+
b
ba
(x
–
b
+
2)(x
(x
+
–
2)
2)(x
=
x
–
2)
–2=2 =x2x–2
– 4= x2 – 4
difference
difference
of
of (a + b)(a
(a +–b)(a
b) =–ab)–=b a – b
2
2
22
2
2 2
2
or
(x
–
or
2)(x
(x
+
–
2)
2)(x
=
x
+
–2
2)
=
=
x
x
–2
–
42 = x(2a–=4x , b( =
a=
2 )x , b = 2 )
=
a
–
b
=
a
–
b
does not
if (amatter
– b) comes
if (a – first
b) comes first
squares
squares It does notItmatter
2
2
2 2
(a + b)(a
=+(ab)+2 =
b)(a
(a + b)(a
b) + b)
(x + 3)(x
=+x3)
+=
2∙xx·2 +3 2∙
+ 3x·2 3 + 32
2
2
= a2 + =
aba+
+baab+ +b2ba + b2
= x2 + =
6xx+
+9 6x + 9
2
= a2 +=2ab
a2 +
+ b2ab
+ b2
2
2
2
2 2
(a – b)(a=–(ab)– b)(a
= (a – b)(a
b) – b)
(x – 4)(x
=–x4)
– 2∙
= xx·2 –4 2∙
+ 4x·2 4 + 42
squaresquare
of
of (a – b)(a2 –= b)
2
a2 –=2ab
a2 –+ 2ab
b2 + b2
= a2 –ab
= a–2 ba
–ab+ –b2ba + b2
= x2 – =
8xx+
–168x + 16
perfecttrinomial
square trinomial
difference
difference A perfectAsquare
= a2 –=2ab
a2 +
– 2ab
b2 + b2
(a + b)(a2 +
= b)
a22 +=2ab
a2 ++ 2ab
b2 + b2
squaresquare
of sumof sum
A perfectAsquare
perfecttrinomial
square trinomial
•
Special
• Special
binomial
binomial
products:
products:
special
special
formsforms
of binomial
of binomial
products
products
that are
thatworth
are worth
memorizing.
memorizing.
•
2
Memory
aid: aid:
(a ± b)
(a22=± (a
2ab
± +2ab
b2)+ b2)
• Memory
(a2 ±= b)
Example:
Example:
Find Find
the following
the following
products.
products.
a
1.
2.
3.
4.
5.
b
22
2
(3y
1. +(3y
4)(3y
+ 4)(3y
– 4) =– (3y)
4) =2 (3y)
–4 –4
= 9y2=– 9y
162 – 16
𝟏𝟏 2 𝟏𝟏 2
b)b=2 a2 – b2
(a + b) (a –
+ b) =
(aa–2 –
1
𝟏𝟏
𝟏𝟏
𝟏𝟏
= 25t=2 +25t
5t2 ++ 𝟒𝟒5t + 𝟒𝟒
2𝟏𝟏
1
1
1
𝟏𝟏
= 9q2=–9q
q 2p–+q𝟑𝟑𝟑𝟑p 𝒑𝒑+2 𝟑𝟑𝟑𝟑 𝒑𝒑2
2
= +a2b)+2 2ab
= a2++b2ab
+ b2
(a + b)2(a
21
2
(t4.+ 𝟏𝟏)
(t 3+=𝟏𝟏)
(t3+=1)
(t2+(t1)
+ 21)
(t + 1)
2
2
= (t =+ (t
2t ++ 1)
2t (t+ +
1)1)
(t + 1)
3
23
22
2
= t += tt +
+ 2t
t ++2t2t ++ t2t+ +1 t + 1
3
32
= t += 3t
t ++3t3t2 ++ 13t + 1
2 2
(2A
5. –(2A
3 +–4B)(2A
3 + 4B)(2A
– 3 ––4B)
3 –=4B)
(2A=–(2A
3)2 –– 3)
(4B)
– (4B)2
a b
b a
a b
Using
• Using
function
function
notation:
notation:
1
a = 5t a ,= 5tb = ,
2 2
2
(3q
3. – (3q
𝒑𝒑)– 𝟔𝟔=𝒑𝒑)(3q)
= (3q)
– 2(3q)�
– 2(3q)�
𝑝𝑝� +6 𝑝𝑝�
� 6+𝑝𝑝�� 6 𝑝𝑝�
𝟔𝟔
6
𝟏𝟏
a = 3y a,= 3y
b = ,4 b = 4
11 2 1 2
2
�𝟓𝟓𝒕𝒕
2. +�𝟓𝟓𝒕𝒕
� +=𝟐𝟐�(5t)
=2 (5t)
+ 2(5t)�
+ 2(5t)�
� + �2�2�+ �2�
𝟐𝟐
2
a
•
ab
2
b=
1
2
2
= –a2b)–2 2ab
= a2+–b2ab
+ b2
(a – b)2(a
1
1
a = 3q a, =b3q
= ,𝑝𝑝 b = 𝑝𝑝
6
amm= a n+ m
anam = a nn+
6
2
(a + b)2(a
=+
a2b)
+22ab
= a2++b2ab
+ b2
Distribute
Distribute
Combine
Combine
like terms.
like terms.
2
- b2A
: –a 3,
= b2A=–4B
3, b = 4B
(a + b) (a –
+ b) =
(aa–2 -b)b2=:aa2 =
2
b =2(2A)
= (2A)
– 2(2A)∙3
– 2(2A)∙3
+ 32 –+16B
32 –2 16B2
2
2
= –a2b)–2 2ab
= a2+–b2ab
: a += b2A,
: ab = 2A,
3 b=3
(a – b)2(a
2
= 4A2=–4A
12A
– +12A
9 –+16B
9 –216B2
Simplify
Simplify
Example:
Example:
GivenGiven
f (x) =f (x)
-3x=+-3x
x2 ,+find
x2 , and
findsimplify
and simplify
1. f 1.
(u –f 1)
(u ,–and
1) , and
2. f 2.
(a +f h)
(a –+ fh)(a)– .f (a) .
1.
2.
f1.(u –f 1)
(u =– -3(u
1) = –-3(u
1) +– (u
1) –+ 1)
(u2 – 1)2
= -3u=+-3u
1 ++u12 –+2u
u2 –+ 2u
1 +1
= u2-=5u
u2+- 5u
2 +2
2
f2.(a +f h)
(a –+ fh)
(a)– =f (a)
[-3(a
= [-3(a
+ h) ++ (a
h) ++ h)
(a2]+–h)(-3a
] – +(-3a
a2)+ a2)
2
= -3a=–-3a
3h –+ 3h
a2 ++ 2ah
a2 ++2ah
h2 + 3a
h2 +
– a3a
– a2
= h2 +
= 2ah
h2 + –2ah
3h – 3h
© 2014 The Critical Thinking Co.™ • www.CriticalThinking.com • 800-458-4849
ReplaceReplace
x with (u
x with
– 1) (u – 1)
2
2
(a – b)2(a
=–
a2b)
–2ab
= a+2 –b2ab
+ b2
Combine
Combine
like terms.
like terms.
ReplaceReplace
x with (a
x with
+ h) and
(a + h)
a. and a.
RemoveRemove
parentheses.
parentheses.
Combine
Combine
like terms.
like terms.
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