Chapter 10
Polynomials and Factoring
3x4 – 11x3 + 0x2 + 1x – 9
3, – 11 , 0 ,1, and – 9
0
constant
monomial
1
linear
binomial
2
quadratic
binomial
3
4
cubic
quartic
polynomial
trinomial
x5 – 2x4 + 4x3 + 23x2
–1
+
+
12x4 + x3 + x2
-
– 3x – 3
2x6 -x5
+3x3 -14x2 +13
7x5 -x4 +9x3 +13x2 +2
2x6 +6x5-x4 +12x3 -x2 +15
-x3 -5x2 +x -1
x3 -3x2 -10x +9
-8x2 -9x +8
5,.6932x4 – 183.321x3 + 2171.09x2 – 10,636.6x + 51,547
5,.6932x4 – 183.321x3 + 2171.09x2 – 10,636.6x + 51,547
x
5
x 2 + 7x – 5x – 35
x2 + 2x – 35
7x2 + x – 35x – 5
7x2 – 34x – 5
-x+4
x2-11x+8
-8x+32
11x2-44x
-x3+4x2
-x3+15x2-52x+32
2x3
9x2
4x3 - 2x 5- 18x2 + 9x4 - 22x + 11x3
-2x5 + 9x4 + 15x3 - 18x2 - 22x
11x
Length of pool
And walkway
A
Width of pool
and walkway
5x + 6
2x + 4
(5x + 6)(2x + 4)
10x2 + 20x + 12x + 24
10x2 + 32x + 24
1344
4664
9984
17,304 26,624
13x + 26 - x3 - 2x2
-x3 - 2x2 + 13x + 26
11x2(-5x+1) +7x(-5x+1) – 3(-5x+1)
-55x3 + 11x2 - 35x2+7 + 15x - 3
-55x3 - 24x2 + 22x - 3
(2x+2)(3x+2)
6x2 + 4x + 6x + 4
6x2 + 10x + 4
X(in)
2
3
4
5
A(in2)
48
88
140
204
a2-b2
(9w)2-32
81w2-9
a2+ 2ab +b2
(12x)2 + 2(12x)(4) + 42
144x2 + 96x + 16
a2- 2ab + b2
(3k)2 -2(3k)(2m)+ (2m)2
9k2- 12km + 4m2
(50-3)(50+3)
2500-9
2491
(60+3)2
3600+360+9
3969
area of entire
square
area of white
region
A
(3x + 2)2
(2x + 3)(2x - 3)
(3x + 2)2 – (2x + 3)(2x - 3)
(9x2 + 12x + 4)-(4x2 - 9)
9x2 + 12x + 4- 4x2 - 9
5x2 + 12x + 13
0.5
0.5
0.5
0.5
0.5
0.5G
0.5G
0.25G2
0.5Gg
25
50
25
0.5
0.5g
0.25g2
0.5g
121m2 -22m + 22m - 4
(9c-1)(9c-1)
81c2 - 9c - 9c + 1
121m2 - 4
81c2 - 18c + 1
992= (100-1)2
=1002 - 2∙100∙1 + 12
=1000 – 200 + 1
=9801
Entire region = (x + 5)2
White region = (x + 3)(x - 3)
Blue region = entire – white
X2 + 10x + 25 - (x2 - 9)
X2 + 10x + 25 - x2 + 9
10x + 34
R (round)
W(wrinkled)
R
RR
RW
W
RW
WW
¼ = 25%
x + 17
x - 12
0
0
-17
0
0
12
-17
12
x-9
9
0
3
7
0
0
1
2
0
-5
0
0
3
7
1
2
-5
3x - 7 = 0
X =7
3
x +1=0
x = -1
x+2=0
x = -2
4x + 9 = 0
X = 4
9
4
7
Solutions are: , -2,
9
3
3
-7
3  (7) 4

 2
2
2
(- 2 - 3)(-2 + 7) = -25
-2
25
-3
3
(0, 5)
6
5
x + 6=0
x = -6
x + 1=0
x = -1
•The x intercepts are x = -8, x = 8
•x coordinate of the vertex is the average of the x intercepts:
8 - 8= 0
2
•Substitute to find the y coordinate
5
y 
(0  8)(0  8)  20
16
•The vertex is (0, 20)
•The arch is 8 - (-8), or 16 feet wide at the base and 20 feet high
sum
product
12
negative
5
p+q
12
pq
5
-16
39
-16
13
3
39
-13
3
different signs
-12
-64
(x - 16)(x + 4)
-16
4
different signs
10
-75
(x + 15)(x - 5)
15
-5
-8
16
1
12
1
-8
1
9
12
is
can
92 - 4(1)(-1)
85
is not
cannot
-1
-
(x -12)( x + 6)
(x - 12)
(x + 6)
x - 12
12
x+6
-6
12
-6
10=(x + 6)(x + 3) -6·3
10= x2 + 9x + 18 -18
0=x2 + 9x -10
0=(x +10)(x -1)
x + 10 = 0 or x – 1 = 0
x = -10 or x = 1
-10
1
(x + 7)(x - 6)=0
x=-7,6
X2 -17x -38
(x + 2)(x - 19) x = -2, 19
1
1
3
7
3
10
21
1
7
3
1
3
22
7
(1x + 7)(3x + 1)
3
3
14
3 3
5
14
5
Multiply 3·5= 15
Products= 15 ·1, 3·5
Sums/Difference= 15 + 1= 16, 15-1=14
3 + 5= 8, 3-5=-2
Which numbers’ product is 15 and sum is 14?
3x2 + 15x -1x - 5
(3x2 + 15x) + (-x - 5)
3x(x + 5) – 1(x + 5)
3(3x - 1)(x + 5)
Put in standard form: 28n2 + 54n + 18 = 0
Factor out 2: 2(14n2 + 27n + 9)=0
Multiply 14 x 9 = 126
Products: 1 x 126 , 2 x 63, 3 x 42, 6 x 21, 9 x 14
Sums: 1+126=127, 2+63=65, 3+42= 45, 6+21=27, 9+14= 25
Which numbers’ product is 126 and sum is 27?
Rewrite problem : (14n2 + 21n + 6n + 9) = 0
Group: (14n2 + 21n)+(6n + 9)=0
7n(2n + 3)+ 3(2n + 3)=0
2(7n + 3)(2n + 3)=0
Answer: 7n+3=0, 2n+3=0
3 3
n=  , 
7 2
28n2
54n
2 14n2
27n
2 7n + 3)
7n + 3
18
3

7
9
2n + 3)
2n + 3
3

2
3

7
3

2
-32
-16
60
60
100
100
-16
60
100
-4(4t2 - 15t - 25)
-4
-4(4t + 5)(t - 5)
4t + 5
t-5
-1.25
5
-1.25
5
5
5
Multiply 28 x 12= 336
Find Products: 2 x 168, 3 x 112, 4 x 84, 6 x 56,
7 x 48, 8 x 42, 12 x 28, 16 x 21
Sums: Instead of finding ALL the sums, which numbers
Are capable of adding or subtracting to equal -5?
12+28=40, 12-28=-16, 16+21=37, 16-21= -5
Rewrite problem: 28x2+16x-21x-12=0
Group: (28x2+16x) +(-21x-12)=0
Factor: 4x(7x+4)-3(7x+4)=0
Rewrite: (4x-3)(7x+4)=0
Factor out a 2: 2(6n2-13n+6)=0
Multiply 6 x 6= 36
Find Products: 1 x 36, 2 x 18, 3 x 12, 4 x 9, 6 x 6
Sums: Instead of finding ALL the sums, which numbers
Are capable of adding or subtracting to equal -13?
9+4= 13 but we need a -13
How about -9-4= -13?
Rewrite problem: 6n2-9n-4n+6
Group: (6n2-9n) + (-4n+6)
Factor: 3n(2n-3)-2(2n-3)=0
Final factor form: 2(3n-2)(2n-3)=0 don’t forget the 2 we factored
At the beginning of the problem.
Rewrite in standard form: 33x2-51x+18=0
Factor out a 3: 3(11x2-17x +6)=0
Multiply 11 x 6 = 66
Find the Products: 1 x 66, 2 x 33, 3 x 22, 6 x 11
Sums: Find a product whose sum is -17
-11-6=-17
Rewrite: 11x2-11x-6x+6=0
Group: (11x2-11x) +(-6x+6)=0
Factor: 11x(x-1) -6(x-1)=0
3(11x-6)(x-1)=0
Set each problem equal to 0
11x-6=0, x-1=0
x= 1, 6/11
Sf = ½ gt2+vit+si
Si= initial position
Sf= final position, g = -32 feet per second
Vi= initial velocity, t = time
0 = ½ (-32)t 2 +140t+200
4( 4t(t-10)+ 5(t-10))=0
0=-16t 2 +140t+200
0= -4(4t 2 -35t-50)
0= -4(4t 2 -40t+5t-50)
-4(4t 2 -40t)+(5t-50)=0
-4(4t+5)(t-10)=0
t?
4t+5=0, t-10=0
t=-5/4, 10
t =10 seconds, why?
n
15
(n+15)(n-15)
11x
12
(11x+12)(11x-12)
3 81
49
3 9p
7
3(9p+7)(9p-7)
x
x+20
3y
3y-2
x 20
20
3y 2
5 C
8
5 c
c
5 c+4
16
4
2
4
(3x)2-52
(3x+5)(3x-5)
3x+5
3x-5
2
1
3
2
1
3
2
1
3
1
2
3
 2 
 10 x 


2
3
4
 
2
3
 2
x

 10
4 

2
 2  3 
2 x  
 10   4 
2
3
x 
10
4
3
3
4
3
3
4
128
0
128
256
-16(1t2-8t +16)
-16(t-4)2
t-4
256
4
4
4
2
2
 4  1
5m 2

  
1  4
1
4
m

m

5
2   5
2 

9x 2  84 x  196
3x  14  0
(3x )2  2(3x )(14)  14 2
2
x 4
3
(3x  14)2
V=256, s=0, h=1024
-16t2+256t+0=1024
-16t2+256t-1024=0 (4t)2-2(4t)(32)+322
16t2-256t+1024
(4t2-32)2=0, t=8 seconds
2·7·x·x·x
7·11·x·x
6·7·x
7·x=7x
7x(2x2-11x-6)
7x(2x+1)(x-6)
10x3(25x2-80x+64)
10x3(5x-8)2
5x
5x(x2+1)
(5x+2)(x2+1)
2x2
2(x2+1)
3x2(4x2+12x+9)
3x2(2x+3)(2x+3)
3x2(2x+3)2
3x3(3x2+90x+5)
(32x3-16x2)+(-98x+49)
16x2(2x -1)+49(-2x+1)
(16x2+49)(2x-1)
(4x+7)(4x-7)(2x-1)
(45x3+9x2)+(5x+1)
9x2(5x+1)+1(5x+1)
(9x2+1)(5x+1)
3x3
3x x2-1
x+1
-1 1
3x+2
x-1
-2/3
2
2 x2-1
x2-1
3x+2
7x
3x2+4x-2
7x
0
0.39
-1.72