1. Factor: 2x3 + 128 2. Solve: 10x3 + 20x2 – x – 2 = 0
3. Divide: x3 – 13x + 8 by x + 4 using
synthetic division. 4. Factor f(x)=2x3 + 7x2 – 33x – 18 given that
f(-6) is a zero. Now find the zeros. 5. Find all the zeros of the function: function f(x) = 2x3 + 3x2 – 39x – 20. Algebra II
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Using the Fundamental
Theorem of Algebra
Algebra II
Algebra II
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1. How many zeros does the function f(x) = x4 + 6x3+ 12x2 + 8x have? 2. How many solutions does the equations x3 + 3x7 + 16x + 48 have? 3. How many roots does f(x) = -3x5 + 4x3 have? Algebra II
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1. 2x3 – 2 = 0
2. x3 + x2 + 7x – 9 = 0
± p/q: ± 1, 3, 9
2(x3 – 1) = 0
1
2(x – 1)(x2 + 1x + 1)=0
x =1
x = (-1 ± i√3)/2
1 1
1 7 -9
2
9 0
x = 1 x2 + 2x + 9 = 0
x = -1 ± 2i√2
Algebra II
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3. x3+3x2+16x+48=0
(x3+3x2)+(16x+48)=0
x2(x+3)+16(x+3)=0
(x + 3)(x2 + 16) = 0
x+3=0 x2 + 16= 0
x=-3
x2 = –16
x = ± 4i
Algebra II
4. x4+6x3+12x2+8x = 0
x(x3+6x2+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 x2 + 4x + 4 = 0
(x + 2)(x + 2) = 0
x = -2 x = -2 6
5. x4 – 14x2 + 49 = 0
(x2 – 7)(x2 – 7) = 0
6. x3 + x2 – 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 x2 = 7 x2 = 7
x = ±√7 x = ±√7
x = -3
Algebra II
x2 – 2x + 5 = 0
x = 1 ± 2i
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f(x) = x4 – 3x3 + 6x2 + 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
x3 – 5x2 + 16x – 30 1. 
Algebra II
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x3 – 5x2 + 16x – 30; x = -2
± p/q: ± 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Algebra II
1 -5 16 -30
-2 1 -7 30 -90
3 1 -2 10 0
x2 – 2x + 10
9
x2 – 2x + 10; x = -2, x = 3
x = 2 ± √(4 – 4(1)(10))
2
x = 2 ± √-36
2 Algebra II
x = 1 ± 3i
{1 ± 3i, -2, 3}
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2. f(x) = x3 – 3x2 – 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
x2 – 8x + 25 Algebra II
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x2 – 8x + 25 ; x = -5
x = 8 ± √(64 – 4(1)(25))
2
x = 8 ± √-36
2 x = 4 ± 3i
Algebra II
{4 ± 3i, -5}
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3. 
g(x) = x4 – 48x2 – 49 0 = (x2 – 49)(x2 + 1)
x2 – 49 = 0 x2 + 1 = 0 x2 = 49
x2 = -1 x=±7
x=±i Algebra II
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4. 
f(x) = -5x3 + 9x2 – 18x – 4 ± p/q: ± 1, 2, 4, 1/5, 2/5, 4/5
-5 9 -18 -4
-1/5 -5 10 -20 0 -5x2 + 10x – 20
-5(x2 – 2x + 4)
Algebra II
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-5(x2 – 2x + 4); x = -1/5
x = 2 ± √(4 – 4(1)(4))
2 Algebra II
x = 2 ± √-12
2 x = 1 ± i√3i
{-1/5, 1 ± i√3i}
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1. 4, -4, and 1
x = 4 x = -4
2. 1, -4, 5
x = 1
(x – 4)(x + 4)(x – 1)
(x2
– 16)(x – 1)
f(x) = x3–x2–16x+16
Algebra II
x = 1 x = -4 x = 5
(x – 1)(x + 4)(x – 5) (x2 + 3x – 4)(x – 5) f(x)=x3–5x2+3x2–15x–4x+20
f(x) = x3 – 2x2 – 19x + 20
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3. -3, 4i
4. 8, -i
x = -3, x = 4i, x = -4i
**imaginary zeros always
come in conjugate pairs!!
(x + 3)(x – 4i)(x + 4i)
*do the imaginary first!
(x – 8)(x + i)(x – i)
(x – 8)(x2 – i2)
(x – 8)(x2 + 1)
(x + 3)(x2 – 16i2)
*remember i2 is -1!
(x + 3)(x2 + 16)
x = 8, x = -i, x = i
f(x) = x3 – 8x2 + 1x – 8
f(x) = x3 + 3x2 + 16x + 48
Algebra II
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5. -3, 2 + i
x = -3, x = 2 + i, x = 2 – i
(x + 3)[(x – 2)2 + 1)
**imaginary zeros always
come in conjugate pairs!!
(x + 3)[x2 – 4x + 4 + 1]
(x + 3)(x – 2 – i)(x – 2 + i)
(x + 3)(x2 – 2x + 5)
*do the imaginary first!
x3 – 2x2 + 5x + 3x2 – 6x + 15
(x + 3)[(x – 2) – i ] [(x – 2) + i]
f(x) = x3 + x2 – x + 15
(x + 3)[(x – 2)2 – i2]
*remember i2 is -1!
(x + 3)[(x – 2)2 + 1]
Algebra II
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6. 2, 5 – i
x = 2, x = 5 – i, x = 5 + i
(x – 2)[(x – 5)2 + 1)
**imaginary zeros always
come in conjugate pairs!!
(x – 2)[x2 – 10x + 25 + 1]
(x – 2)(x – 5 + i)(x – 5 – i)
(x – 2)(x2 – 10x + 26)
*do the imaginary first!
x3 – 10x2 + 26x – 2x2 + 20x – 52
(x – 2)[(x – 5) + i ] [(x – 5) – i]
f(x) = x3 – 12x2 + 46x – 52 (x – 2)[(x – 5)2 – i2]
*remember i2 is -1!
(x – 2)[(x – 5)2 + 1]
Algebra II
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5. -4, 1, 7
Algebra II
6. 10, -√5
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7. 8, 3 – i Algebra II
Omit 8!!!
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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)=0.00547x3 − 0.225x2 + 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
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2. Between 1985 through 1995, the number of home
computers, in thousands, sold in Canada is estimated by
this equation c(t) = 0.92(t3 + 8t2 + 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
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