Solving Polynomial Functions
We already know how to solve linear and quadratic equations.
We usually do this to find a function’s x-intercepts. This can
also be done with higher degree functions.
f (x)
x 3 2x 2 15x
0
x 3 2x 2 15x
0
x(x 2 2x 15)
0
x(x 5)(x 3)
x
0, x
5, x
3
Solving Polynomial Functions
Factor the following expression
2x 5 18x 3
2x 3 (x 2
9)
2x 3 (x 3)(x
3)
Solving Polynomial Functions
Solve the following equation…
0
4 x 4 16x 3 16x 2
0
4 x 2 (x 2
0
4 x 2 (x 2)(x 2)
x
0, x
4 x 4)
2
Solving Polynomial Functions
There are special factoring patterns for cubics that you
can use to factor them.
The sum of two cubes:
a3 b3
(a b)(a3
The difference of two cubes:
a3
ab b3 )
b3
(a b)(a3
ab b3 )
Solving Polynomial Functions
An example using the sum of cubes pattern.
The sum of two cubes:
a3 b3
(a b)(a2
Factor:
ab b2 )
x 3 64
x3
43
(x 4)(x 2
4 x 42)
(x 4)(x 2
4 x 16)
Solving Polynomial Functions
An example using the sum of cubes pattern.
The difference of two cubes:
a3
b3
(a b)(a2
Factor:
x3
27
x3
33
(x
3)(x 2
(x
3)(x 2
ab b2 )
3x 32 )
3x 9)
Factoring by Grouping
This is an appropriate factoring method when there are
exactly four terms in the polynomial.
Factor the following expression:
x3
3x 2 16x 48
3)
)
(xx 2 (x
3)( 3) 16 (x
( x 3)(x 4)(x 4)
Practice
Solve the following expression:
0
x3
0
x 2 (x 7) 9(x 7)
0
7x 2
9x 63
(x 7)(x 2
9)
0 (x 7)(x 3)(x 3)
x
7, x
3, x
3
Practice
Solve the following expression:
0 16x 4
0 (4 x 2
0
625
25)(4 x 2 25)
(2x 5)(2x 5)(4 x 2 25)
5
5
5
x
,x
,x
i
2
2
2
Practice
Solve the following expression:
2x5 24x 14x3
0 2x5 7 x3 24x
0
0
2 x( x
4
2 x( x
x 0, x
2
7x
2
3)( x
3, x
12)
2
4)
4
3
3x
12
x2
x4
4x
x2
4
Practice
The city of Encinitas is impressed with your math work on the
firework display, so they commission you to design a marble
basin that will go in front of the Cardiff kook. The sides and
bottom should be 1 foot thick and the outer length should be
twice the outer width and height. They want the basin to hold
36 cubic feet of water.
1 foot
36 (2x 2)(x 2)(x 1)
x
0 2x3 8x 2 10x 40
0
x
2
2 x ( x 4) 10( x 4)
0 ( x 4)(2 x 2 10)
2x
Divide Polynomials
If You know one of the factors of a number. You can
divide it through to find the other.
2049
10 20490
20
04
0
49
40
90
90
0
Divide Polynomials
If You know one of the factors of a number. You can
divide it through to find the other. There might be a
remainder
3
10
2049
10 20493
20
04
0
49
40
93
90
3
Divide Polynomials
If You know one of the factors of a polynomial. You
can divide it through to find the rest.
x2 2x 2 2x4
2x4
Solution : 2 x 2 7 x 10
2x 2
3x 3 0 x 2
4 x3 4 x 2
3
2
7 x 4x
3
2
7 x 14x
7 x 10
5x 1
5x
14x
10x 2 9x 1
10x 2 20x 20
11x 21
11x 21
x
2
2x 2
Divide Polynomials
If You know one of the factors of a polynomial. You
can divide it through to find the rest.
3x 2 4x 3
x 2 3x 5 3x 4 5 x 3 0 x 2 4 x 6
3 x 4 9 x 3 15x 2
3
2
4x 15x 4x
3
2
4 x 12x 20x
2
3x 16x 6
3x 2 9 x 15
25x 9
2
25x 9
Solution : 3x 4 x 3 2
x 3x 5
Divide Polynomials
Divide x 3 2 x 2
3
x 3 x
x3
x2
2
2x
3x 2
5x 2
5x 2
5x
6
21x 18 by ( x 3)
Is ( x 3) a factor of the polynomial?
21x 18
Express the polynomial as a product of
its factors.
21x
15x
6x 18
6x 18
0
x3 2x 2 21x 18
( x 2 5 x 6)( x 3)
( x 6)(x 1)(x 3)
Divide Polynomials
- 3 is a solution to x 3 4 x 2 11x 30 0
What are the other solutions?
3
x 3 x
x3
x2
2
4x
3x 2
7 x2
7x2
7x
10
11x 30
Express the polynomial as a product of
its factors.
0 x3 4x2 11x 30
11x
21x
10x 30
10x 30
0
0
( x 2 7 x 10)( x 3)
0 ( x 2)(x 5)(x 3)
x 2, x 5, x
3
Divide Polynomials
2 is a zero of f ( x) 2 x 3 2 x 2 8 x 8
What are the other zeros?
2x 2
3
2
x 2 2x 2x
2 x3 4 x 2
6x2
6x2
6x
4
8x 8
Express the polynomial as a product of
its factors.
0 2 x 3 2 x 2 8x 8
8x
12x
4x 8
4x 8
0
0 (2 x 2 6 x 4)( x 2)
0 (2x 2)(x 2)(x 2)
x
1, x
2, x 2
Divide Polynomials
- 3 is a zero of f ( x) 2 x 3 3x 2 23x 42
What are the other zeros?
x 3 2 x3
3
2x
2x 2 3x 14
3x 2 23x 42
2
6x
2
3x 2 23x
3x 9 x
14x 42
14x 42
0
Express the polynomial as a product of
its factors.
0 2x3 3x2 23x 42
0
(2 x 2 3 x 14)( x 3)
0 (2x 7)(x 2)(x 3)
x
7
,x
2
2, x
3