pre-calculus mathematics 12 - 2.1 - polynomials 1. define ...divide a polynomial function using long...

Pre-Calculus Mathematics 12 - 2.1 - Polynomials

Definitions:

A polynomial expression has the form:

and

1. exponents must be whole numbers (may skip descending powers)

2. coefficients must be real numbers

The polynomials degree is defined by n and its leading coefficient is a.

Example 1:

Polynomial Degree Leading coefficient Name

( )

( ) √ x

( )

( )

( )

Shape: Polynomial functions are continuous, no breaks or sharp corners

Goals: 1. Define and recognise properties of a polynomial function 2. Determine the real zeros and their multiplicity given a polynomial function


The End Behaviour of polynomial functions is determined by the leading coefficient

End Behaviour

Domain

Range

n

a

n basic shape of the graph; a the direction(s) end behaviours

Turning points – generated by the ‘middle’ terms of a polynomial function.

If the degree is n, the maximum number of turning points is n–1 ( but could be fewer).

Example 2: Determine the degree, the end behaviour and the maximum number of turning points of the

following polynomial functions.

a) ( ) √ b) ( )

Zero(s) of a Polynomial Function – location(s) where the polynomial function crosses the x-axis

a) also called root(s), solution(s), x-intercept(s)

b) determined by factoring a polynomial expression

c) zeros can be real or imaginary (we only deal with real in PC 12)


Example 3: Find all the real zeros of ( )

A polynomial function of degree n has, at most, n real zeros.

Remember, the zeros of the polynomial are really just the x-intercepts.

For leading term, , if n is even, may cross the x-axis from 0 to n times.

Eg: ( )

For leading term, , if n is odd, may cross the x-axis from 1 to n times. (must cross at least once)

Eg: ( ) √ x

Multiplicity – A polynomial of degree n can have at most n distinct solutions. When a solution is

repeated r times, the solution has a multiplicity of r.

Example 4: Find all the real zeros, and the multiplicity of each zero.

a) ( ) b) ( ) ( )

Practice: Pg 73-76 # 2- 5, 6(a-f), 7(a,c,e,g,i), 9(c,d,h,i,l,n)

Pre-Calculus Mathematics 12 - 2.2 – Graphing Polynomial Functions

Graphing a polynomial function can be summarized using the following steps:

1. Locate the zeros from factoring. (2.1)

2. Locate the y-intercept. (set x = 0 and solve for y)

3. Determine the shape at each zero from the multiplicities. (See below)

Multiplicity 1 ( ) n is Even ( ) n is Odd and >1

Factor ( ) ( ) ( ) ( ) ( )

Graph shape

at Zero

4. Determine the end behaviour using . ( n even/odd; a > 0 or a < 0)

5. Estimate the highs and lows of the function using x values between the zeros.

(Substitute x-values into the original function)

6. Plot a reasonable number of points and draw a smooth continuous curve.

Example 1: Sketch: ( ) ( ) ( )

Goals: 1. Sketch the graph of a given polynomial function 2. Sketch the graph of a polynomial function given knowledge about the function


Example 2: Determine the equation of the polynomial ( ) in factored form, if ( ) and

the zeros are 0, 0, –1,

Example 3: Determine the equation of the polynomial in factored form.

Example 4: Sketch a graph of a polynomial function of lowest degree.

Value of f(x) – + + –

Zeros of x

–3 –1 2


Example 5: Sketch: ( ) ( ) ( )( )

Practice: Pg 81- 83 #1(a,d,f),2 (a, b), 3(a, b),5(a, b) ,6 (a),7 (a, b)

Pre-Calculus Mathematics 12 - 2.3 – Division of Polynomials

Dividing a polynomial function using long division uses the same techniques as long division of numbers.

Recall: 1537 12

We can also divide polynomial expressions using these same techniques

The result of the division of a polynomial P(x) by a binomial D(x) can be written in the forms:

( ) ( ) ( ) ( )

( ) ( )

( ) where Q(x) is the quotient and R is the

remainder (if it exists)

Example 1:

Divide: 3 27 14 8 1x x x x

and write final solution is both forms.

Example 2:

Divide: 3 25 10 13 9 2x x x x

and write final solution is both forms.

Goals: 1. Divide a polynomial function using long division and synthetic division.

2. Write the results in both forms of division statements:

( ) ( ) ( ) ( )

( ) ( )

( ).

3. Determine unknown values in a polynomial equation given divisor(s) and remainder(s).


Example 3: Divide: 4 3 2 22 3 7 12 4 4x x x x x and write final solution is both forms.

An alternate (and much quicker and easier) method of dividing a polynomial by a binomial is through the

use of synthetic division. It performs the division without writing variables and reduces the amount of

calculations.

Example 4: Divide the following using synthetic division. 3 25 10 13 9 2x x x x

Example 5: Divide the following using synthetic division. 3 22 3 5 2 2 1x x x x

Example 6: Divide using synthetic division: 4 3 2 22 3 7 12 4 4x x x x x and write final solution is both

forms.

Practice: Pg 89 #1(a,c,d,h)


NOTE: In order to do both synthetic division and long division, all descending powers of the polynomial

P(x) must be present. If any power is missing from P(x), consider it as a coefficient of zero. ( ex 0x2 )

Example 7: Divide the following: 34 7 5 3x x x

Synthetic division can also be used as a quick technique to determine a missing value of a polynomial.

Example 8: When 4 33 1x x kx is divided by 2x the remainder is – 5.

Practice: Pg 90

#2(a,c,f,h,k),

3(b,c,d,g,h)

Practice: Pg 92 #4(a-f)

Pre-Calculus Mathematics 12 - 2.4 – The Remainder and Factor Theorems

The Remainder Theorem When a polynomial ( )P x is divided by a binomial in the form x a , the remainder is ( )P a

Example 1:

Determine the remainder when 3 23 5 15 9x x x is divided by 1x Example 2:

Determine the remainder when 3 24 8 5 3x x x is divided by 2 1x

Example 3:

Determine k if 3 24 3 1 2y ky y y has a remainder of 1.

Goals: 1. Use the remainder theorem to determine the remainder of a polynomial

2. Use the factor theorem to find factors of a polynomials


Example 4: When 3 22 8kx x mx is divided by 2x the remainder is – 38. When it is divided by 1x

the remainder is 7. Find k and m.

The Factor Theorem When a polynomial ( )P x has a factor of x a if and only if ( ) 0.P a

Example 5: Is 1x a factor of 3 2( ) 3 3P x x x x

Example 6: Is 2 1x a factor of 3 2( ) 8 6 9 5P x x x x

Practice: Pages 97 to 99 # 1, 2 (a ,b,c,d),3 (a ,b ,c,e)


The Rational Root Theorem

If 1 2( ) ...n n nf x ax bx cx jx k is a polynomial function with integer coefficients, every zero of f(x)

is of the form m

n where m is the possible factors of k and n is the possible factors of a.

Example 7: Factor fully 4 3 25 2 20 24x x x x


Example 8: Factor fully 3 2 2 5 4 3x x x

Practice: Pages 98 to 99 # 4 (a – f), 5 & 6 below

Pre‐Calculus Mathematics 12 ‐ 2.5 – Polynomial Applications

Goals: 1. Solve real‐live problems involving polynomials applying all the techniques learned throughout this chapter.

Definitions:

When solving any problem involving a polynomial functions:

1. Draw a picture. This helps organize the data given and clarify the question.

2. Establish your unknown, and create a polynomial equation (usually cubic) from the information.

3. Determine all the solution (roots) of the polynomial.

4. Reject any solution(s) that are not possible in the problem.

5. Answer the question with a sentence.

Example 1:

A storage crate has a height that is twice the width, and a length that is 8m longer than the width. If the

volume is 80m3, find the dimensions of the crate.

Pre‐Calculus Mathematics 12 ‐ 2.5 – Polynomial Applications

Example 2:

An open box is made from a piece of cardboard measuring 30 cm x 20 cm. Cutting out squares from each

corner and folding the edges up makes a box with a volume of 1056 cm3. Determine the height of the box.

Example 3:

A silo is the shape of a cylinder topped by a hemisphere. The overall height of the silo is 24 ft. Find the radius if the

volume is 1440π ft3

Practice: Pg 102 # 1, 2, 3, 4, 5, 6, 8

pre-calculus mathematics 12 - 2.1 - polynomials 1. define ...divide a polynomial function using long...

Documents