Download 5-5 Theorems About Roots Teacher Notes

WARM-UP
Multiply: x  a
5-5 THEOREMS ABOUT ROOTS OF
POLYNOMIAL EQUATIONS
Ms. Miller
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CONJUGATES
Complex Conjugates: a  bi and a  bi
4i
Example: Find the conjugate of 4  i
Find the conjugate of
6i
6i
Irrational Number Conjugates: a  b and a  b
Example: Find the conjugate of 2  3 2  3
Find the conjugate of
7
 7
CONJUGATE ROOT THEOREM
If P( x) is a polynomial with rational coefficients, then irrational roots
that have the form a  b occur in conjugate pairs. So, if a  b is an
irrational root, then a  b is a root.
If P( x) is a polynomial with rational coefficients, then irrational roots
that have the form a  bi occur in conjugate pairs. So, if a  bi is an
irrational root, then a  bi is a root.
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USING THE CONJUGATE ROOT THEOREM
1. A cubic polynomial P(x) has rational coefficients. If 3  2i and
are two roots of P(x)=0, what is one additional root?
5
2
3  2i
2. A cubic polynomial P(x) has rational coefficients. If 2  3i and
are two roots of P(x)=0, what is one additional root?
2
3
2  3i
USING THE CONJUGATE ROOT THEOREM
What is a quartic polynomial equation that has roots 2-3i, 8, 2?
1) Check the degree of
the polynomial. Find
missing roots.
2) Write linear factors.
3) Multiply.
P( x)  ( x  8)( x  2)( x  (2  3i ))( x  (2  3i ))
P( x)  ( x 2  10 x  16)( x 2  x(2  3i )  x(2  3i )  (2  3i )(2  3i ))
P( x)  ( x 2  10 x  16)( x 2  2 x  13)
P( x)  ( x 2  10 x  16)( x 2  2 x  13)
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FUNDAMENTAL THEOREM OF ALGEBRA
If P ( x) is a polynomial of degree n  1 , then P ( x )  0 has exactly n
roots including multiple and complex roots.
• A polynomial with no constant term has 0 as one of its roots
Has a GCF of x, so x = 0 is a root
f ( x)  4 x3  x
• A polynomial with even degree doesn’t have to cross x-axis
No real solutions
f ( x)  x 2  2 x  2
• A polynomial with odd degree must cross x-axis
End behavior is either up & down or down & up
• A polynomial with odd degree must have root in set of real numbers
USING FUNDAMENTAL THEOREM OF ALGEBRA
What are the zeros of P ( x)  2 x 4  3 x 3  x  6 ?
1. Graph the function & find any real roots: x  1, 2
2. Use synthetic division until you have a quadratic function:
2 2 -3 0 -1 -6
1 2 1 2 3
4
2 1
2
2
6
0
4
3
2 1 3
2 1 3 0
3. Use Quadratic Formula to find complex roots of 2 x 2  x  3
x
1  1  4(2)(3)
4

3  23
2

3  23
2
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