5-1 Operations with Polynomials

Monomial – Degree of a monomial – Identify the degree of each monomial.

a) 8x b) 23 c) 43r s d) 2 3x yz

Simplified when (a) there are no power of powers, (2) each base appears exactly once, (3) all fractions are simplified, and (4) there are no negative exponents.

Simplify each expression.

1) 5 2 3c c c 2) 3 2 4 1a a b c

3) 3 52gh g h

4)

7

3 5

24

3

wz

w z 5)

5 3

7

6

18

s x

sx

6)

2

10

n

n

7) 6

3z 8)

23

4

3a

b

9)

23

2 5

2xy

x y

Property Definition Example

Product of Powers a b a bx x x

Quotient of Powers a

a b

b

xx

x

Negative Exponent 1 1a a

a ax and x

x x

Power of a Power b

a a bx x

Power of a Product a a axy x y

Power of a Quotient

a a aa

a

x x x yand

y y y x

Zero power 0 1x

Polynomial Degree of polynomial – Determine if each expression is a polynomial. If it is a polynomial, state the degree of the polynomial.

10) 5 2 7316

4p p t 11) 4 4 18c c 12) 2 13 7x x

Simplify each expression by adding or subtracting the polynomials.

13) 2 3 2(15 3 11) (2 6 1) x x x x x 14) 2 2(3 5 ) ( 4 ) x x x x

15) 3 2 3(2 9 ) (5 4 7 ) x x x x x 16) 2 2(3 ) ( 6 ) x x x

Simplify by using the distributive property.

17) 23 (2 5 )x y x 18) 2 24(6 9 12)

3x x x 19) 2 3 22 (6 14 30 4) r r r r

20) 2(3 1)(2 4 5)x x x 21) (2 3)( 1)( 4)x x x

5-2 Dividing Polynomials Divide a polynomial by a Monomial.

1) 2 3 3 45 15 10

5

a b ab a b

ab 2) 2 4 3 1(9 3 27 )(3 ) xy x y xy xy

Arithmetic Long Division - Perform the following division problems using arithmetic long division. Then identify the divisor, quotient, dividend, and remainder. 3) 945 9 4) 277 12 5) 4572 8 Divide by using long division

6) 2(15 8 12) (3 1) x x x 7) 2(20 13 2) (4 1) x x x

Divide by using long division.

9) 2 1(6 7 6)(3 2) x x x 9) 2 1( 3 20 12)( 6) x x x

10) 3 2(2 10 5) ( 5) x x x x 11) 4( 5 10) ( 3) x x x

5-2 Dividing Polynomials SYNTHETIC DIVISION- **the divisor must be a LINEAR binomial in the form (x - a)

Example 1: 2( 2 24) ( 4) z z z

Constant of x r

( 4r in our example)

4 1 2 -24 Degree of terms in descending order

Answer:____________________________

(**Start with the power of one degree less.)

Example 2: 3 2(5 13 10 8) ( 2) x x x x

Constant of x r

( 2r in our example)

2 5 -13 10 -8 Degree of terms in descending order

Answer:____________________________

Divide by using synthetic division.

1)

2(6 5 6) ( 3) x x x 2) 4 3(3 7 5 1) ( 2) x x x x

3) 2( 2 3) ( 5) x x x 4) 4 3( 3 7 14) ( 4) x x x x

Divide the polynomials using the method indicated: Long division Synthetic division

7) 2(12 36 15) (6 3) y y y 8) 5 2( 3 20) ( 2) z z z

Application

9) An experimental electrical system has a voltage that can be modeled by 3 2( ) 0.5 4.5 4 , V t t t t where t represents time in seconds. The resistance in the system also

varies and can be modeled by ( ) 1. R t t The current I is related to voltage and resistance by

the equation .V

IR

Write an expression that represents the current in the system.

10) If the remainder of polynomial division is 0, what does it mean?

5-3 Polynomial Functions

Polynomial in one variable –

Degree of a polynomial –

Leading coefficient – Standard form (of a polynomial) – Standard Form

Leading Coefficient: Degree: No. of Terms: Name:

Rewrite each polynomial in standard from. Then identify the following a) the leading coefficient, b) the degree, c) the number of terms, and d) name the polynomial. If it is not a polynomial in one variable, explain why.

1) 2 32 7 x x 2) 3 2 26 4 a a ab 3) 5 2 63 2 4 8 x x x 4) 38 x

Type No. of Terms Example

Monomial 1 22x

Binomial 2 4x

Trinomial 3 3 1 x x

Polynomial 4 or more 3 23 4 1 x x x

Name Degree Example Name Degree Example

Constant 0 9 Cubic 3 3 22 1 x x x

Linear 1 4x Quartic 4 4 3 22 3 4 1 x x x x

Quadratic 2 2 3 1 x x Quintic 5 5 4 3 27 3 2 1 x x x x x

3 23 5 6 7 x x x

Polynomial function – Find p(-2) and p(3) for each function.

5) 2( ) 7 5 9 p x x x 6) 5 3( ) 4 p x x x 7) 3 2( ) 3 2 5 p x x x x

If 2( ) 3 4 p x x and 2( ) 2 5 1 r x x x , find each value.

8) (8 )p a 9) 2( )r a 10) ( 2)r x

Application: The volume of air in the lungs during a 5-second respiratory cycle can be modeled by

3 2( ) 0.037 0.152 0.173 v t t t t , where v is volume and t is time in seconds. Find the volume of air in

the lungs 1.5 seconds into a respiratory cycle.

Graphs of Polynomial Functions

Constant Degree 0

Linear Degree 0

Quadratic Degree 2

Cubic Degree 3

Quartic Degree 4

Quintic Degree 5

End Behavior – a description of the end value of a function as it approaches positive infinity

(x ∞) and negative infinity (x -∞).

Polynomial End Behavior Chart

P(x) has… Odd Degree Even Degree Leading coefficient: 0a

As ,x ( )P x

As ,x ( )P x

As ,x ( )P x

As ,x ( )P x

Leading coefficient: 0a

As ,x ( )P x

As ,x ( )P x

As ,x ( )P x

As ,x ( )P x

Identify the leading coefficient, degree, and end behavior.

11) 5 2( ) 2 3 4 1 P x x x x 12) 2( ) 3 1 S x x x 13) 7 4( ) 4 5 T x x x

LC: ___________ LC: ___________ LC: ___________

Degree: _________ Degree: _________ Degree: _________

End Behavior: End Behavior: End Behavior:

As ,x ( )P x As ,x ( )S x As ,x ( )T x

As ,x ( )P x As ,x ( )S x As ,x ( )T x

Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient.

14) 15) 16) 17)

5-4 Analyzing Graphs of Polynomial

Functions

Location Principle - 1) Make a table and determine consecutive integer values of x between which each real zero of the

function 4 3 2( ) 4 1f x x x x is located. Then draw the graph.

x 4 3 24 1x x x f(x)

Relative Maximum - Relative Minimum – Turning Points -

2) Make a table for 3 2( ) 3 5f x x x . Estimate the x-coordinates at which the relative maxima

and relative minima occur. Then draw the graph.

x 3 23 5x x f(x)

Steps for Graphing a Polynomial Function

a.

b.

c.

d.

e.

f.

3) Graph the polynomial function, 3 2( ) 3 3 f x x x , by making a table of values.

x 3 23 3 x x f(x)

As ,x ( )f x As ,x ( )f x

4) Graph the polynomial function, 4 2( ) 7 10 f x x x , by making a table of values.

x 4 27 10 x x f(x)

As ,x ( )f x As ,x ( )f x

5) Graph the polynomial function, 3 2( ) 2 5 5 f x x x x , by making a table of values.

x 3 22 5 5 x x x f(x)

As ,x ( )f x

As ,x ( )f x

6) Graph the polynomial function, 4 2( ) 6 5 f x x x , by making a table of values.

x 4 26 5 x x f(x)

As ,x ( )f x

As ,x ( )f x

5.5 Factoring Polynomials Factor Theorem – When factoring a polynomial, always look first for a _________. 2 Terms 3 Terms 4 Terms Difference of Two Squares Perfect Square Trinomial Grouping Sum of Two Cubes General Trinomial Difference of Two Cubes Quadratic Like Trinomial

Sum of Two Cubes: 3 3 2 2( )( ) x y x y x xy y Ex: 3 28 ( 2)( 2 4) x x x x

Difference of Two Cubes: 3 3 2 2( )( ) x y x y x xy y Ex: 3 227 ( 3)( 3 9) x x x x

Factoring by Grouping

1) 3 27 3 21 x x x 2) 3 2 25 25 z z z 3) 2 3 3 3 2 3 3 33 2 3 2 x y xy y x z xz z

Factoring the Sum or Difference of Two Cubes

4) 3 64x 5) 3 125y 6) 68m

Quadratic Form (quadratic like).

7) 4 26 8 x x 8) 5 33 54 x x x 9) 4 81x

Prime polynomials - When a polynomial cannot be factored it is prime. 10) Identify any prime polynomials.

3 400x 4 32 1 x x 6 664 x y

Solve by Factoring!

11) 38 343 0 x 12) 327 8 0 x 13) 3 125 0 r

14) 3 28 8 0 x x x 15) 4 25 4 0 x x 16) 4 16 0 x

17) 4 3 24 3 12 0 s s s s 18) 3 25 2 10 0 x x x 19) 3 24 9 36 0 x x x

5-6 The Remainder and Factor Theorems

Remainder Theorem – Synthetic Substitution-

1) 4 3If ( ) 3 2 5 2, find (4)f x x x x f .

(Method 1) Synthetic Substitution (Method 2) Direct Substitution Use synthetic substitution to find p(-2) and p(4) for each function.

2) 4 3( ) 2 5 4 6 p x x x x 3) 2( ) 2 5 3 P x x x

4) The function 3 2C(x)=2.46x 22.37 53.81 548.24x x can be used to approximate the number,

in thousands, of international students studying in the United States x years SINCE 2000. How many international college students can be expected to study in the U.S. in 2015? Depressed Polynomial – Factor Theorem –

5) Determine whether 3x is a factor of 3 24 15 18 x x x . Then find the remaining factors of the polynomial. Given a polynomial and one of its factors, find the remaining factors of the polynomial.

6) 3 22 5 6; x-1 x x x 7) 3 22 25 12; x+3 x x x

5-7/5-8 Fundamental Theorem of Algebra

Rational/Irrational Root Theorem The following statements are equivalent:

A real number is a of the polynomial equation .

______________

r is an of the graph of .

is a factor of .

When you the rule for by , the remainder is .

is a of .

The Fundamental Theorem of Algebra – Corollary: State the number of zeros including real, complex and repeated roots for each equation.

1) 3 23 4 12 0 x x x 2) 4 256 0 x 3) 2 6 9 0 x x Write the simplest polynomial function with the given zeros.

4) 2,2,4 5) 2

0, ,33

6) 1

2, , 22

Rational Root Theorem –

**p

q= possible rational zeros

Example: 2q

x 3 - 11x 2 +12x + 9p

= 0 so possible rational zeros are as follows

p

q= _____________________ =

Example: Use the rational root theorem to find all possible rational roots

7) 3 23 10 24 0 x x x 8) 3 22 9 16 0 x x x 9) 4 33 5 7 12 0 x x x Irrational Root Theorem – *Irrational roots come in conjugate pairs* To find the real roots: (follow these steps) 1. Use the rational root theorem to identify possible rational roots 2. Test the possible roots to find one that is actually a root.

***You can use synthetic substitution for this. 3. Factor and find the remaining roots. Identify all of the real roots of the following equation.

10) 3 23 4 12 0 x x x List all possible rational zeros here: _____ _____

1 2

_______________ _______________

Roots are:

Find all of the real roots of the function:

11) 3 22 5 4 3 0 x x x 12) 3 26 6 4 0x x x

13) 4 3 22 7 8 12 0x x x x 14) 3 2 10 12 0x x x Complex Conjugate Theorem – Find the missing zeros then write the simplest polynomial function with the given zeros.

15) 4 2i and 16) 1 3 i and 17) 5 3 and

Solve the equations by finding ALL roots.

18) 4 3 24 16 20 0 x x x x

19) 4 3 23 5 27 36 0 x x x x 20) 3 23 4 12 0 x x x Application 21) A grain silo is in the shape of a cylinder with a hemisphere top. The cylinder is 20 feet tall.

The volume of the silo is 2106 cubic feet. Find the radius of the silo.