7/29/2019 Lesson Four Polynomials

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Polynomials

Quintic5

Quartic4

Cubic3

Quadratic2

Linear1

Constant0

ExampleNameDegree

( ) 5f x =

( ) 3 2f x x= +

2( ) 4f x x=

3( ) 2 3f x x x= +

4( ) 3f x x x= +5 3( ) 1f x x x x= + +

Some of the Special Names of the Polynomials of thefirst few degrees:

A polynomial with one term is called a monomial.

Example: ( )f x x=

A polynomial with two terms is called a binomial.

Example: ( ) 1f x x= +

A polynomial with three terms is called a trinomial.

Example:2( ) 1f x x x= + +

Find a quarticfunction with zeros and and3 4

( ) 1( 3)( 3)( 4)( 4)f x x x x x= + +

(2) 60f =

Find the zeros of the function:2 2

( ) ( 16)( 25)g x x x= +

Answers: 4, 5x i=

Answer:

Find a cubic function with zeros 2 +i and 1.

3 2( ) 5 9 5f x x x x= +

Answer:

Find a quarticfunction with zeros and

4 2( ) 2 3f x x x=

Answer:

i 3.

If 4 +i is a root of the polynomial equation below, find the other roots.

4 , 2i Answer:

If the polynomial function below is divisible by z 3 i, find the otherroots.

3 , 11i

Answer:

3 2( ) 6 34f z z z z= + +

3 2( ) 5 56 110f z z z z= + +

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If is a zero of the function find thevalue ofa.

2i 4 2( )h x x x a= + +

Answer: a =-12

Find the equation of a cubic function with x-intercepts at 1, 2 and 3and a y-intercept at -6.

3 2( ) 6 11 6f x x x x= +

Answer:

Use synthetic division to divide:

3 25 5 2x x x+ + by 2x+

Answer:2 3 1 Remainder: 0x x+

Use synthetic division to divide:

3 210 8x x x+ + by 1x+

Answer:2 10 Remainder: 18x

The Remainder Theorem:

When a polynomial P(x) is divided by x a, the

remainder is P(a).

Example:

Find the remainder when the function below is divided by x 1.

20 10( ) 2 4 1f x x x= +

Answer: Remainder is -1

The polynomial below has a remainder of 4 when divided by(x + 1). Calculate the value ofk.

10 2( )g x x x x k= + + +

Answer: k =3

The polynomial below has a remainder of 8 when divided by(x - 1). Calculate the value ofa.

4( ) 3h x x ax= + +

Answer: a =4

The Factor Theorem

For a polynomial P(x), x a is a factor if and only ifP(a) =0.

Example: Below is a polynomial equation and one of its roots, findthe remaining roots.

3 26 11 4 4 0 with a root of 2x x x x+ = =

Answer: Remaining roots are:1 22 3,x =

Example: Below is a polynomial equation and one of its roots, findthe remaining roots.

3 22 5 23 10 0 with a root of 2x x x x+ + = =

Answer: Remaining roots are:12, 5x =

Example: Below is a polynomial equation and one of its roots, findthe remaining roots.

4 3 24 4 25 6 0 with roots of 2 and 3x x x x x x + + = = =

Answer: Remaining roots are:12

x =

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The polynomial below has a factor of (x 1) and a remainder of 8when divided by (x +1). Calculate the values ofa and b.

3 2( ) 2f x x x ax b= + +

Answer: a =-5, b =6

The polynomial is a factor of the polynomial below.Calculate the value ofa.

24 3x x +

3 2( ) ( 4) (3 4 ) 3g x x a x a x= + + +

Answer: a =1

Solve the following polynomials:

3 1 0x =Answer:

1 3 1 3

2 21, ,i ix + =

38x = Answer:

2, 1 3, 1 3x i i= +

330 3x + = Answer:

3 3 3 3 3 3

2 23, ,i ix + =

Solve the following polynomials:

4 24 5x x+ =

Answer:

1, 1,2, 2x =

6 4 24 4 0x x x + = Answer:

1, 1, 2, 2, 2, 2x i i=

5 3 29 8 72x x x + =

Answer:

3, 3, 2,1 3,1 3x i i= +

Rational Root Theorem

1 2

1 2 1 0 0( ) ... , where 0n n

n nP x a x a x a x a x a a

= + + + + +

LetP (x) be a polynomial of degree n with integral coefficients and anon-zero constant term:

If one of the roots of the equation P(x) =0 is where p and qare non-zero integers with no common factor other than 1,

p

qx =

Then p must be a factor of , and q must be a factor of .0a na

Given:4 3 2

3 13 15 4 0x x x+ + =

Find all of the roots using the rational zero test.

1 13 1 13

6 62, 2, ,x + =

Given:3 2

2 5 6 2 0x x x+ + + =

Find all of the roots using the rational zero test.

12, 1 , 1x i i= +