1. Describe the end behavior of the graph y = 2x5 – 3x2 + 5.

2. Sketch a graph of 3rd degree with a zero at -5 (multiplicity 2) and a zero at 0 (multiplicity 1).

1

Real Zeros of Polynomial Functions.

To solve, you factor the problem, each factor must be factored down to at most a quadratic.

To solve, put each factor equal to zero.

3

1. 2x5 + 24x = 14x3

2x5 – 14x3 + 24x = 0

2x(x4 – 7x2 + 12) = 0

2x(x2 – 3)(x2 – 4) = 0

2x=0 x2–3=0 x2–4=0

x=0 x2=3 x2= 4

x=±√3 x = ±2

2. 2y5 – 18y = 0

2y(y4 – 9) = 0

2y(y2 – 3)(y2 + 3) = 0

2y=0 y2–3=0 y2+3=0

y=0 y2 = 3 y2 = –3

y=±√3 y=±i√3

4

1. Long Division

2. Synthetic Division

5

Divide 3x3 – 5x2 + 10x – 3 by 3x + 1

3x + 1 ) 3x3 – 5x2 + 10x – 3

6Algebra II

x2

3x3 + x2

-6x2+ 10x

- 2x

-6x2 - 2x

12x - 3

+ 4

12x + 4

-7

-7 . 3x + 1

+

Divide 2x3 – 7x2 – 17x – 3 by 2x + 3

2x + 3 ) 2x3 – 7x2 – 17x – 3

7

x2

2x3 + 3x2

-10x2

- 17x

- 5x

-10x2 - 15x

- 2x - 3

- 1

-2x - 3

0

Divide 6x2 + x – 7 by 2x + 3

2x + 3 ) 6x2 + x – 7

8

3x

6x2 + 9x

-8x- 7

- 4

- 8x - 12

5

5 . 2x + 3+

Divide 4x3 + 5x2 - 3x + 10 by x2 + 3

x2 + 0x + 3 ) 4x3 + 5x2 – 3x + 10

9

4x

4x3 + 0x2 + 12x

5x2 – 15x + 10

+ 5

5X2 + 0x + 15

-15x – 5

- 15x – 5 x2 + 3

Divide -3x3 + 4x – 1 by x - 1

x - 1 ) -3x3 + 0x2 + 4x – 1

10

-3x2

-3x3+ 3x2

-3x2+ 4x

- 3x

-3x2 + 3x

1x - 1

+ 1

1x - 1

0

Divide a polynomial by a binomial

The polynomial must be in standard form

Be sure to add zeros when needed for missing placeholders

The divisor must be a linear binomial with a leading coefficient of 1 and in the form x + k

11Algebra II

12

Divide x3 + 2x2 – 3x - 6 by x + 2.

X = -2

-2 1 2 - 3 -6

times

1 0

0

-3

6

Now solve x2 – 3 = 0 to find the remaining zeros.

- + +2

13

Divide 6x3 – 19x2 + 16x - 4 by x - 2.

X = 2

2 6 -19 16 -4

times 12

6 -7

-14

2

4

0

Now solve 6x2 – 7x + 2 = 0

+ + +

14

15

V = x3 + 9x2 – x – 105

Given sides: x – 3 and x + 5

16Algebra II

1. Divide 6x3 + 11x2 – 12x – 13 by 3x + 1 using long division.

2. Divide 10x3 – 10x2 – 2x + 4 by x + 3 using synthetic division.

3. Factor 4x3 – 3x2 – 25x – 6 using synthetic division given that one zero is -2.

4. Solve 6x3 + 23x2 – 33x + 10 given that one zero is -5.

17Algebra II