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