1. Factor: 2x 3 + 128
2. Solve: 10x 3 + 20x 2 – x – 2 = 0
3. Divide: x 3 – 13x + 8 by x + 4 using
synthetic division.
4. Factor f(x)=2x 3 + 7x 2 – 33x – 18 given that
f(-6) is a zero. Now find the zeros.
5. Find all the zeros of the function:
function f(x) = 2x 3 + 3x 2 – 39x – 20.
Algebra II 1

Algebra II
Using the Fundamental
Theorem of Algebra

Algebra II 3

1. How many zeros does the function
f(x) = x 4 + 6x 3 + 12x 2 + 8x have?
2. How many solutions does the equations
x 3 + 3x 7 + 16x + 48 have?
3. How many roots does f(x) = -3x 5 + 4x 3
have?
Algebra II 4

1. 2x 3 – 2 = 0
2(x 3 – 1) = 0
2(x – 1)(x 2 + 1x + 1)=0
x =1 x = (-1 ± i √ 3)/2
2. x 3 + x 2 + 7x – 9 = 0
± p/q: ± 1, 3, 9
1 1 7 -9
1 1 2 9 0
Algebra II x = 1 x 2 + 2x + 9 = 0
x = -1 ± 2i √ 2
5

3. x 3 +3x 2 +16x+48=0
(x 3 +3x 2 )+(16x+48)=0
x 2 (x+3)+16(x+3)=0
(x + 3)(x 2 + 16) = 0 x+3=0 x 2 + 16= 0
x=-3 x 2 = –16
x = ± 4i
Algebra II
4. x 4 +6x 3 +12x 2 +8x = 0
x(x 3 +6x 2 +12x+8)= 0
± p/q: ±1,2,4,8
1 6 12 8
1 1 7 19 27
-1 1 5 7 1
2 1 8 28 64
-2 1 4 4 0 x = 0 x = -2 x 2 + 4x + 4 = 0
(x + 2)(x + 2) = 0
x = -2 x = -2
6

5. x 4 – 14x 2 + 49 = 0
(x 2 – 7)(x 2 – 7) = 0
x 2 = 7 x 2 = 7
x = ± √ 7 x = ± √ 7
Algebra II
6. x 3 + x 2 – x + 15 = 0
± p/q: ± 1,3,5,15
1 1 -1 15
1 1 2 1 16
-1 1 0 -1 16
3 1 4 11 48
-3 1 -2 5 0 x = -3 x 2 – 2x + 5 = 0
x = 1 ± 2i
7

1.
f(x) = x 4 – 3x 3 + 6x 2 + 2x – 60
± p/q: ± 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
1 -3 6 2 -60
1 1 -2 4 6 -54
-1 1 -4 10 -8 -52
2 1 -1 4 10 -40
-2 1 -5 16 -30 0
x 3 – 5x 2 + 16x – 30
Algebra II 8

x 3 – 5x 2 + 16x – 30; x = -2
± p/q: ± 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
1 -5 16 -30
-2 1 -7 30 -90
3 1 -2 10 0
x 2 – 2x + 10
Algebra II 9

x 2 – 2x + 10; x = -2, x = 3
x = 2 ± √ (4 – 4(1)(10))
2
x = 2 ± √ -36
2
x = 1 ± 3i
{1 ± 3i, -2, 3}
Algebra II 10

2. f(x) = x 3 – 3x 2 – 15x + 125
± p/q: ± 1, 5, 25, 125
1 -3 -15 125
1 1 -2 -17 108
-1 1 -4 -11 136
5 1 2 -5 100
-5 1 -8 25 0
x 2 – 8x + 25
Algebra II 11

x 2 – 8x + 25 ; x = -5
x = 8 ± √ (64 – 4(1)(25))
2
x = 8 ± √ -36
2
x = 4 ± 3i
{4 ± 3i, -5}
Algebra II 12

3.
g(x) = x 4 – 48x 2 – 49
0 = (x 2 – 49)(x 2 + 1)
x 2 – 49 = 0 x 2 + 1 = 0
x 2 = 49 x 2 = -1
x = ± 7 x = ± i
Algebra II 13

4.
f(x) = -5x 3 + 9x 2 – 18x – 4
± p/q: ± 1, 2, 4, 1/5, 2/5, 4/5
-5 9 -18 -4
-1/5 -5 10 -20 0
-5x 2 + 10x – 20
-5(x 2 – 2x + 4)
Algebra II 14

-5(x 2 – 2x + 4); x = -1/5
x = 2 ± √ (4 – 4(1)(4))
2
x = 2 ± √ -12
2
x = 1 ± i √ 3i
{-1/5, 1 ± i √ 3i}
Algebra II 15

1. 4, -4, and 1
x = 4 x = -4 x = 1
(x – 4)(x + 4)(x – 1)
(x 2 – 16)(x – 1)
f(x) = x 3 –x 2 –16x+16
2. 1, -4, 5
x = 1 x = -4 x = 5
(x – 1)(x + 4)(x – 5)
(x 2 + 3x – 4)(x – 5) f(x)=x 3 –5x 2 +3x 2 –15x–4x+20 f(x) = x 3 – 2x 2 – 19x + 20
Algebra II 16

3. -3, 4i
x = -3, x = 4i, x = -4i
**imaginary zeros always come in conjugate pairs!!
(x + 3)(x – 4i)(x + 4i)
*do the imaginary first!
(x + 3)(x 2 – 16i 2 )
*remember i 2 is -1!
(x + 3)(x 2 + 16) f(x) = x 3 + 3x 2 + 16x + 48
Algebra II
4. 8, -i x = 8, x = -i, x = i
(x – 8)(x + i)(x – i)
(x – 8)(x 2 – i 2 )
(x – 8)(x 2 + 1) f(x) = x 3 – 8x 2 + 1x – 8
17

5. -3, 2 + i
x = -3, x = 2 + i, x = 2 – i
**imaginary zeros always come in conjugate pairs!!
(x + 3)(x – 2 – i)(x – 2 + i)
*do the imaginary first!
(x + 3)[(x – 2) – i ] [(x – 2) + i]
(x + 3)[(x – 2) 2 – i 2 ]
*remember i 2 is -1!
(x + 3)[(x – 2) 2 + 1]
(x + 3)[(x – 2) 2 + 1)
(x + 3)[x 2 – 4x + 4 + 1]
(x + 3)(x 2 – 2x + 5) x 3 – 2x 2 + 5x + 3x 2 – 6x + 15 f(x) = x 3 + x 2 – x + 15
Algebra II 18

6. 2, 5 – i
x = 2, x = 5 – i, x = 5 + i
**imaginary zeros always come in conjugate pairs!!
(x – 2)(x – 5 + i)(x – 5 – i)
*do the imaginary first!
(x – 2)[(x – 5) + i ] [(x – 5) – i]
(x – 2)[(x – 5) 2 – i 2 ]
*remember i 2 is -1!
(x – 2)[(x – 5) 2 + 1]
Algebra II
(x – 2)[(x – 5) 2 + 1)
(x – 2)[x 2 – 10x + 25 + 1]
(x – 2)(x 2 – 10x + 26) x 3 – 10x 2 + 26x – 2x 2 + 20x – 52 f(x) = x 3 – 12x 2 + 46x – 52
19

5. -4, 1, 7 6. 10, √ 5
Algebra II 20

7. 8, 3 – i Omit 8!!!
Algebra II 21

1. A tachometer measures the speed (in revolutions per minute, or RPMs) at which an engine shaft rotates. For a certain boat, the speed X (in hundreds of RPMs) of the engine shaft and the speed
S (in miles per hour) of the boat are modeled by s(x)=0.00547x
3 − 0.225x
2 + 3.62x − 11.0. a)   What is the tachometer reading when the boat travels 15 miles per hour?
b)   What is the speed of a boat that shows a tachometer of 2000
RPMs
Algebra II 22

2. Between 1985 through 1995, the number of home computers, in thousands, sold in Canada is estimated by this equation c(t) = 0.92(t 3 + 8t 2 + 40t + 400), where t is the number of years since 1985. How many computers where there in 1992? In what year did home computer sales reach 2 million?
Algebra II 23