Alg 4 Rational Roots
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3.6 Synthetic Division
I
Find f(x) = x3 x 2
x 6
x 1 using synthetic division
Coefficients of
Numerator
zero of
divisor
1 1
1
-1
-6
1
remainder
II Examples: Use synthetic division to find the quotient and remainder, please.
1) 3x 4 x3 x 3
x 1
2) Now subst ( -1) for x
f(x) = 3x 4 + x 3 - x + 3
f(- 1) =
x 3 - 4x 2 + 6x - 4
3)
x-2
Alg 4 Rational Roots
4) Can’t use synthetic division so use either long division or factoring to find all the roots
x 4 3x 2 4
x2 1
5) Find all the roots using synthetic division if -3 and 2 are two of the roots for the following:
f(x) = x4 + 4x3 x2
16x
12
2
Alg 4 Rational Roots
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III Writing polynomial equations
1) factors are 2x 1 ,
2) factors are 2i, 1 3i
3x 7
Alg 4 Rational Roots
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Algebra 3 Assignment # 4
(1) Divide each of the following please.
(a)
x 4 + 4x 3 + 3x 2
(b)
7x 4
(c)
2x 4 + 5x3 + 50x + 12
x + 4
(d)
6x3 + 13x 2
3x
(e) 10x
(f)
x5
14x 3
3
4x
4
6x 2 + 17x
6
17x + 4
3x 2 + 2x + 8
1
x + 2
x
x
2
1
2x + 1
1
(2) Use the given root to find the remaining roots for each of the following please.
3
2
(a) 2x + 13x + 9x
(b) 2x
3
5x 2
4
8x + 6 , given
3
(c) x + 4x + 3x
4
(d) 2x + 3x
3
30 , given
2
4x
29x 2
5 is a root.
3
2
is a root.
4 , given
2 is a root .
18x + 72 , given 3 and
2 are roots .
(3) Write a polynomial having 1 , 2 , and 4 as roots.
(4) Let f x = 2x
4
3x 3
5x 2 + cx
3 . Find the value of c if f 3 = 45 .
Alg 4 Rational Roots
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Algebra 3 Assignment # 4
Answers
(1)
3
(a) x + 2x
(b) 7x
(c) 2x
3
3
2
4
2
2
4
6x + 5 +
x
2
4
3x 2 + 12x + 2 +
(d) 2x 2 + 5x
(e) 5x
x
4
5
2x + 1
4x + 3 +
3
x
2
(f) x + x + x + x + 1
(2)
(a) the roots are
5,
(b) the roots are
3
2
(c) the roots are
2,
(d) the roots are
(3) x3 + x2
(4) c = 4
4,
10x + 8
3
33
4
,2
2
2,
1
2,3,
3
2
4
Alg 4 Rational Roots
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CREATING EQUATIONS
Roots = zeros = x - intercepts
Given roots
r1 and r2
-b
a
Sum
x r1 x r2
product=
x 2 x 4
x2
2x 8
0
x2
0
1
2
ex.2 x= ,
4
3
4x=1 3x=2
4x-1 3x 2
12x 2
11x 2
c
a
b
2
c
8
a
1
a
1
a 1 b=2 c=-8
Given roots
ex.1
0
2x 8
0
b 1 2 3+8
11
=
=
a
4 3
12
12
c
2
=
a 12
a = 12 b = -11 c = 2
0
0
12x 2 -11x+2=0
ex.3 x = 2 i , 2 i
x 2
x 2
i
2
x 2
i2
x2
4x
x2
4x 5
0
4 1 0
0
i
0
b
=
a
c
=
a
a=1
x2
4
1
4-i2
5
=
1
1
b = -4 c = 5
4x 5
0
Alg 4 Rational Roots
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Creating Equations Assignment
Find the polynomial equations determined by the following roots.
1)
1 2
,
4 3
5
2) 3 i , 2 i
1) 12x 3 55x 2 27x 10 0
2)
x 4 10x 3 39x 2 70x 50 0
Alg 4 Rational Roots
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3.7 Roots of Polynomials
I The Remainder Theorem
If a polynomial p(x) is divided by ( x – c ) , the remainder is equal to p(c).
Ex.
If you divide by c (use synthetic division) , then the remainder is = p(c)
x2
If p(x) =
2 1
8
10
2
1 10
20
30
8x
10 and you divide by ( x – 2 ) then p (2) = the remainder
then
2
p(2) = 2 + 8(2) +10 = 30
II The Factor Theorem
A polynomial p(x) has a factor ( x – c ) if and only if p ( c ) = 0
2
Ex.
If f(x) = x
5x + 6 and f( 2 ) = 0, then ( x – 2 ) is a factor of f ( x )
f(2) = 22 - 5 2 + 6 = 0
III Examples
1) Is (x – 2) a factor of f (x) x 5 16x 2 3
What is f(2)?
2) Is x 1 a factor of g(x) x 4
6? What is g( 1 ) ?
5x 2 + 10x
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ROOTS = ZEROS = X-INTERCEPTS
III Finding the roots of polynomials
If 2x 3 4x 5
8x 2 2x 15 , what are the roots of the polynomial?
Look at the roots and look at the first and last terms of the polynomial. Any conclusions?
Examples: Find all the roots of the polynomials, please.
1) f x
x3 3x 2 10x 24
2) g x
2x 4 3x3 6x 2 12x 8
Alg 4 Rational Roots
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Algebra 3 Assignment # 5
Find all roots of each of the following please.
(1) 2x 3
3x 2
8x
3
0
(2) x 4
5x 3
4x 2
8x
5
4x 3
x2
4
(3) x
4
(4) 3x
(5) 12x
4
10x 3
8
0
0
6x 2
2x
1
0
x2
4x
1
0
12
0
0
12x 3
(6) x
4
x3
13x 2
(7) x
3
4x
3
(8) x
3
4x 2
6x
4
(9) x
3
7x 2
17x
15
(10) x
4
6x 3
12x 2
6x
13
(11) x
3
6x 2
21x
26
0
(12) 4x
(13) 8x
4
(14) 2x
(15) x
4
4
4
8x 3
25x
0
0
33x 2
4x3
0
72x
6x 2
27
5x
x3
2x 2
19x
2x 3
4x
4
0
1
10
0
0
0 ; Given 1 + i is a root
(16) x 4
4x 3
5x 2
16x
4
(17) x 4
2x 3
7x 2
30x
50
0 ; Given 2i is a root
0 ; Given
2 + i is a root
(18) Write an integral polynomial having the following sets of roots please
(a)
2,
3,
3
4
(b)
1
2i ,
2
3
Alg 4 Rational Roots
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Algebra 3 Assignment # 5
Answers
(1)
3,1, 1
2
(3)
2,1,2,
1
3i
2
(4) 1 , 1 , 1
3
i
(10)
5
2
1,3
2i
(12) 1 , 3 ,
2 2
3i
3
(13) 1 , 1 , 1 , 1
2 2 2
(14)
2, 1 ,1
2
2i
2
(16)
2i ,
2
3
(15) 1
i,
(17) 2
i,1
3i
3
2
(18) (a) 4x + x
1
(8) 2 , 1 i
2
2, 2
2, 1,
(6) 3 , 1 , 1 , 4
13
(9) 3 , 2
(11)
1
3
3
(5) 1 , 1 ,
2 2
(7) 1 ,
(2)
27x + 18 = 0
4
(b) x + 2x
3
2x 2 + 18x + 5 = 0
Alg 4 Rational Roots
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Algebra 3 Assignment # 6
Review Worksheet
(1) Find all roots of each of the following please.
(a) 6x 4
(b) 4x
(c) x
3
x3
55x 2
20x 2
x
9x
5
9
0
0
4
3x 3
x2
4
2x 3
7x 2
4x4
37x3 + 37x2 + 9x
(d) x
(e) 4x5
7x
62x
30
0
52
0 , Given: 2 + 3i is a root
9 = 0
(f) x5 + 6x4 + 15x3 + 26x2 + 36x + 24 = 0
(2) Write an integral polynomial having the following sets of roots please.
(a)
1,
2 3
,
3 5
(3) Find the remainder when the polynomial 7x103
(b)
2
5x53 + 6x16
5 ,
3
i
4 is divided by x
i .
Alg 4 Rational Roots
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Algebra 3 Assignment # 6
Answers
(1)
3, 1 ,
(a)
(b)
1
2
5,
3, 2 ,
(c)
(d) 2
3i ,
1
2
(e) 1 ,
(f)
(2)
2,
(a) 15x
,
2,
3
12i
1 2i
3
5
3
2,
14x 2
(b) x4 + 2x3
(3) 2
1
3
2
3i
7x
15x2
6
46x
0
10
0