W11D1Conjugate Root Theorem
Warm Up
1. Given that the roots of a polynomial are x=3, 6, -5 what is the polynomial? x3 − 4x2 − 27x + 90
2. Multiply (3+4i)(3-4i)
25
√
√
3. Multiply (9- 2)(9+ 2)
79
Lesson 29 Complex Roots of Polynomials and Graphs
Add a column for them to sketch in notes. Will do that on Wednesday.
1.
2.
3.
4.
5.
F unction
x3 − x2 − 2x
−2x4 + 2x2
−2x3 + 8x2 + 10x
x5 + x3 + 8x2 + 8
x6 − 10 ∗ x2 + 9
N umberof roots
3
3
3
1
4
N umberof Extrema
2
3
2
2
3
Sketch
Degree = n
Number of roots = n
Number of min and max (extrema) = n-1
n includes imaginary roots
n includes roots with a multiplicity greater than 1
MULTIPLICITY is having the same root appear more than once. It happens when the factor has an exponent. On the graph
it looks like the function barely touching the x axis at that value.
Graph the parabola y=x2 − 6x + 9 = (x-3)(x-3)
Roots; x=3, x=3
When you solve for the roots you only get one on the graph, but it is still a 2nd degree polynomial.
EX 1:
Find all the roots of f (x) = x3 − 14x2 + 69x − 116 given (x-4) is a factor
x2 − 10x + 29
x−4
3
x − 14x2 + 69x − 116
− x3 + 4x2
− 10x2 + 69x
10x2 − 40x
29x − 116
− 29x + 116
0
1
4
1
x = 4, 5±2i
− 14
69
− 116
4
− 40
116
− 10
29
0
EX 2:
Find all roots of
f (x) = x3 − 4x2 + 2x + 4 given that x=2 is a root
x2 − 2x − 2
x−2
x3 − 4x2 + 2x + 4
− x3 + 2x2
− 2x2 + 2x
2x2 − 4x
− 2x + 4
2x − 4
0
1
−4
2
4
2
−4
−4
−2
−2
0
2
1
x2 − 2x − 2 = 0
(x − 2)2 √
=2+1
x=2± 3
x = 2, 1 ±
√
3
Can you make an observation about the roots?
EX 1:
has roots that are complex conjugates. EX 2:
has roots that are irrational conjugates (radical)
Conj. Root Theorem: If you know one complex or irrational root of a polynomial, and all coefficients are real and rational,
the conjugate is also a root. Think of warm up - multiply them and get a real number.
EX 5:
Find all roots of f (x) = x4 − 3x3 + 6x2 + 2x − 60 given that 1+3i is a root.
What are the factors?
What is another root? 1-3i
(x − (1 − 3i))(x − (1 + 3i))
(1)
(x − 1 + 3i))(x − 1 − 3i))
(2)
Make a 3 by 3 box OR group the terms
(3)
((x − 1) + 3i)((x − 1) − 3i)
(4)
Difference of squares
(5)
(x − 1)2 − (3i)2
(6)
2
x − 2x + 1 + 9
(7)
2
x − 2x + 10
(8)
Then long division to find other roots. Can’t use synthetic unless it is degree = 1.
x2
2
x − 2x + 10
4
−x −6
3
x − 3x + 6x2 + 2x − 60
− x4 + 2x3 − 10x2
− x3 − 4x2 + 2x
x3 − 2x2 + 10x
− 6x2 + 12x − 60
6x2 − 12x + 60
0
x2 − x − 6 = 0
(x − 3)(x + 2) = 0
x = 3, −2
x = 1 ± 3i, 3, -2
EX 6:
Find all roots of f (x) = x4 − 6x3 + 30x2 + 24x − 136 given that 3+5i is a root
(x − 3 + 5i)(x − 3 − 5i)
(x − 3)2 − (5i)2
x2 − 6x + 9 − 25i2
x2 − 6x + 34
x2
2
x − 6x + 34
4
3
−4
2
x − 6x + 30x + 24x − 136
− x4 + 6x3 − 34x2
− 4x2 + 24x − 136
4x2 − 24x + 136
0
x=3±5i, ± 2
EX 7:
One of the roots of f (x) = x3 − 5x2 + 23x − 51 is 1+4i. Find the other roots.
(x − 1 + 4i)(x − 1 − 4i)
(x − 1)2 − (4i)2
x2 − 2x + 1 − 16i2
x2 − 2x + 17
x −3
2
x − 2x + 17
3
2
x − 5x + 23x − 51
− x3 + 2x2 − 17x
− 3x2 + 6x − 51
3x2 − 6x + 51
0
x = 3, 1+4i, 1-4i
EX 8:
Write the equation of the polynomial with roots 3 +
f (x) = (x − 3 +
√
5)(x − 3 −
√
5, 3 and -1
√
5)(x − 3)(x + 1)
2
f (x) = (x − 6x + 9 − 5)(x2 − 2x − 3)
f (x) = (x2 − 6x + 4)(x2 − 2x − 3)
f (x) = x4 − 8x3 + 13x2 + 26x − 12
Assume the leading coefficient is 1; you don’t need to solve for ”a” value in these problems (but you could right)?
Exit Pass
1. Find the roots of the polynomial f(x) = 3x3 − 29x2 + 92x + 34 given that 5+3i is one of the roots.
3x + 1
2
x − 10x + 34
3
2
3x − 29x + 92x + 34
− 3x3 + 30x2 − 102x
x2 − 10x + 34
− x2 + 10x − 34
0
x = 5 ± 3i, − 13
Extra Problems
6.
Find all roots of
f (x) = x3 − 4x2 + 6x − 4 given that 1- i is a root
x2 − 2x + 2
x−2
x3 − 4x2 + 6x − 4
− x3 + 2x2
− 2x2 + 6x
2x2 − 4x
2x − 4
− 2x + 4
0
x = 2, 1+i, 1- i
7. Find all roots of
f (x) = x4 − 6x3 + 30x2 + 24x − 60 given that 1+3i is a root
x2 − 4x + 12
x2 − 2x + 10
x4 − 6x3 + 30x2 + 24x − 60
− x4 + 2x3 − 10x2
− 4x3 + 20x2 + 24x
4x3 − 8x2 + 40x
12x2 + 64x − 60
− 12x2 + 24x − 120
88x − 180
x = 1 ± 3i, 3, -2
8. Find all roots of
f (x) = z 3 + 6z 2 + 61z + 106 given that -2+7i is a root
x
2
x + 4x + 53
3
+2
2
x + 6x + 61x + 106
− x3 − 4x2 − 53x
2x2 + 8x + 106
− 2x2 − 8x − 106
0
Exit Pass
Start HW - it is sort of long
9.
Find all roots of
x3 − 3x2 − 25x − 21 given that x+1 is a factor
x2 − 4x − 21
x+1
3
x − 3x2 − 25x − 21
− x3 − x2
− 4x2 − 25x
4x2 + 4x
− 21x − 21
21x + 21
0
x = -1, 7,3
10.
Find all roots of
x3 − 5x2 + 2x + 8 given that x+1 is a factor
x2 − 6x + 8
x+1
3
x − 5x2 + 2x + 8
− x3 − x2
− 6x2 + 2x
6x2 + 6x
8x + 8
− 8x − 8
0
x = -1, 4, 2
11. Find all complex numbers of the form z = a + bi , where a and b are real numbers such that z z’ = 25 and a + b = 7
where z’ is the complex conjugate of z
2x2 + 3
x2 − 4
2x4 − 5x2 − 12
− 2x4 + 8x2
3x2 − 12
− 3x2 + 12
0
Graphs for W11D3 with the Chart
5
4
3
2
1
−5 −4 −3 −2 −1
−1
−2
−3
−4
−5
−6
−7
−8
−9
5
4
3
2
1
1
2
3
4
5 −5 −4 −3 −2 −1
−1
−2
−3
−4
−5
−6
−7
−8
−9
Figure 1: x3 − x2 − 2x
1
2
3
4
5
3
4
5
− 2x4 + 2x2
15
13
11
9
7
5
3
1
−4
−3
−2
−1−1
−3
−5
1
2
3
19
17
15
13
11
9
7
5
4
3
1
−1
−5 −4 −3 −2 −1
Figure 2: x6 − 10 ∗ x2 + 9
1
2
x5 + x3 + 8 ∗ x2 + 8
46
41
36
31
26
21
16
11
6
1
−5 −4 −3 −2 −1
−4
−9
1
2
3
4
5
6
7
Figure 3: −2x3 + 8x2 + 10x
1.
2.
f (x) = x3 − x2 − 2x
4
f (x) = −2x + 2x
5
3
x(x − 2)(x + 1)
3 roots, 2 extrema
2
− 2x (x − 1)(x + 1)
4 roots, 3 extrema
2
3. f (x) = x + x + 8x + 8
5 roots, 4 extrema
Imaginary Roots Roots are x=-2, x= ± i
4. f (x) = −2x4 + 2x2
x = 0, 2, −1
2
2
x = 0, ±1
3
−∞
−∞
(x + 1)(x + 8)
2x2 (1 − x2 )
4 roots, 3 extrema
−∞
∞
−∞
∞
−∞
−∞
∞
−∞
Multiplicity of Roots! Roots are x=0, x=0, x=1, x=-1
5. f (x) = −2x3 + 8x2 + 10x
3 roots, 2 extrema
x = 3, x = 4
− 2x(x + 1)(x − 5)
2
x − 7x + 12
6.
What polynomial has the roots
7.
What 3rd degree polynomial has the roots
8.
What 4th degree polynomial has the roots x=4, x=-9, x=0 and opens downward with a stretch factor of 4?
x3 − 7x + 2
x = −2, x = −1, x = 3
x(x − 4)(x + 9) = −4x(x − 4)(x + 9)any factor can have multiplicity = −4x3 − 20x2 + 144x
9.
What 3rd degree polynomial has the roots
x = 0, x = 2, x = −1
CHALLENGE What 3rd degree polynomial has the rootsx = 3, x = 2 ±
√
√
(x − 3)(x − 2 + 11)(x − 2 − 11)
√
√
(x − 3)([x − 2] + 11)([x − 2] − 11)
√
(x − 3)((x − 2)2 − ( 11)2 )
10.
√
x3 − x2 − 2x
11
(x − 3)(x2 − 4x + 4 − 11) = (x − 3)(x2 − 4x − 7) = x3 − 7x2 + 5x + 21
In groups W8D3 Polynomial Graphs Worksheet
Exit Pass
1. Find the roots of the function f(x) = 2x4 + 4x2 − 48x x=4, -6, 0
2. Write the equation of the function that has roots at x = ±3, x=-7 x3 + 7x2 − 9x − 63
1. Given that x= 3 is a root find all other roots of 5x3 + 14x2 − 3x
x = 3, 0, 51