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Zeros of Polynomial Functions
Take out your homework
Warm-up1. Solve 3x3 – 2x2 + 3x – 4 ÷ x – 3 by Synthetic Division2. Find f(25) using synthetic substitution in the above
expression3. List all the potential real factors of the expression in #1.4. Quiz Review:
5 4 3 2 25 10 20 13 3 9 4 3x x x x x x x
Warm-up Answers
1.
2. 45,6963. ±1, ±2, ±4, ±4/3,±2/3,
±⅓
4.
2 683 7 24
3x x
x
3 22
110 785 10 5 23
4 3
xx x x
x x
Learning GoalsFundamental Theorem of AlgebraConjugate Root TheoremComplex Zeros…be able to find a polynomial
given zeros, real and complex…be able to factor and find
complex zeros of a polynomial function
Fundamental Theorem of AlgebraAn nth-degree polynomial (n >0),
has at least one zero in the complex number system.
◦Corollary – An nth-degree polynomial function has exactly n zeros, including repeated zeros, in the complex number system.
Conjugate Root TheoremIf a + bi (b ≠ 0), is a root of a
polynomial function with real coefficients, then the complex conjugate, a - bi is also a root of the polynomial.
Find a Polynomial Function Given Its ZerosWrite a polynomial function of
the least degree possible, with real coefficients in standard form with the given zeros.
Pg. 127: #35 (homework problem)
**hint – Conjugate Root Theorem
Solution:
Zeros: -1, 8, (6 – i)
4 3 2
( ) ( 1)( 8)[ (6 )][ (6 )]
19 113 163 296
f x x x x i x i
x x x x
Use Synthetic Division
85.
86.
Practice (10 min.)Pg. 127: #32, 33, 38, 85-86Write the polynomial function in
standard form, given the zeros.
32. 3, -4, 6, -133. -2, -4, -3, 538. 7, 7,4i
4 3( 1) ( 2)x x x
4 3 2(3 2 5 4 2) ( 1)x x x x x
Wrap-up and Quiz ReviewI am able to find a polynomial
given zeros, real and complex.Quiz Friday covering Long
Division and Synthetic Division/Substitution.
Quiz ReviewTake out your Quiz Review
Worksheet.
Factoring PolynomialsEVERY polynomial can be written
as the product of linear factors and/or irreducible quadratic factors, each with real coefficients.
A quadratic factor is irreducible over the reals when it has real coefficients, but no real zeros associated with it. ◦Example: (x2+4)
Factor and Find the Zeros of a Polynomial FunctionStarting point?
◦Total number of Zeros◦Rational Zero Theorem – List all
possible Rational Zeros◦Graph to find likely real zeros
Use Synthetic Division to:◦Check for factors◦factor the function
ExampleFactor and find the zeros of the
equation:
Total zeros – 5RZT – All possible rational zeros
◦– {±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30}Find potential zeros on a graph that
are in the list.
5 3 218 30 19 30x x x x
Example continued:
Choose possible zeros to try with synthetic substitution. ◦(-5, 2, 3 are all good possibilities)
-5 1 0 -18 30 -19 30
-5 25 -35 25 -30
1 -5 7 -5 6 0
2 1 -5 7 -5 6
2 -6 2 -6
1 -3 1 -3 0
3 1 -3 1 -3
3 0 3
1 0 1 0
4 3 25 7 5 6x x x x
3 23 3x x x
𝑥2+1
Factor polynomial – all linear factorsContinue from the previous
Example:
Rewrite all irreducible factors as complex linear factors.
2
2
( 1) 0
1
1
: ( )( )
x
x
x
x i
factors x i x i
( ) ( 5)( 2)( 3)( )( )f x x x x x i x i Final answer written in all linear factors
Wrap-upI am able to factor and find
complex zeros of a polynomial function.
Practice
Pg. 127: #42-44, Complex Zeros WS42-44 – write as a) the product of irreducible linear
factors, b) the product of linear factors, and c) list all the zeros (Real and Complex)
42. 43. 44.
4 3 2( ) 3 12 20 48g x x x x x
4 3 2( ) 3 12 8g x x x x
4 3 2( ) 2 15 18 216h x x x x x
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