1. Write the following polynomial in factored form. Show your work.

y = x^4 - x^3 - 6x^2

2. For the following function state the zeros and determine the multiplicity of any multiple zeros. Show your work.

f (x) = x^4 - 8x^3 + 16x^2

3. Write the following function in standard form. Show your work.

y = (x - 5) (x + 5) (2x - 1)

4. Write a polynomial in standard form with the given zeros. Show your work.

-1, 3, 4

1. y = x^2 (x^-3) (x+2)

2. x=0 multiplicity 2 and x=4 multiplicity 2

3. y = 2x^3 - x^2 - 50x + 25

4. y = x^3 - 6x^2 + 5x+12

Step-by-step explanation:

1. y = x^4 - x^3 - 6x^2

Factor out an x^2

y = x^2 (x^2 - x-6)

We can factor the polynomial inside the parenthesis

What multiplies to - 6 and add to - 1

-3*2 = - 6 and - 3+2 = -1

y = x^2 (x^-3) (x+2)

2. f (x) = x^4 - 8x^3 + 16x^2

We need to factor out an x^3

f (x) = x^2 (x^2 - 8x + 16)

We should notice inside the parentheses is the difference of two cubes

(a^2 - 2ab-b^2) = (a-b) ^2 where a=x and b=-4

f (x) = x^2 (x-4) ^2

We can use the zero product property to find the roots

0 = x^2 (x-4) ^2

x^2 = 0

x = 0 with multiplicity 2 (because it is a square)

(x-4) ^2 = 0

x-4 = 0

x=4 with multiplicity 2 (because it is a square)

3. y = (x - 5) (x + 5) (2x - 1)

We know (x-5) (x+5) is the difference of squares

(a-b) (a+b) = a^2 - b^2 where a=x and b=5

y = (x ^2 - 5^2) (2x - 1)

y = (x ^2 - 25) (2x - 1)

Now we FOIL first outer inner last

y = x^2 * 2x + - 1*x^2 + - 2x*25 - 1*-25

y = 2x^3 - x^2 - 50x + 25

Standard form is from highest exponent to lowest exponent.

4. Using the zero product property we know that

(x-a) (x-b) (x-c) = 0 where a, b, c are the zero's of the function

(x--1) (x-3) (x-4) = 0

(x+1) (x-3) (x-4)

Now FOIL the first two terms

(x^2 - 3x+1x-3) (x-4)

(x^2 - 2x-3) (x-4)

Now we multiply x by the first term then multiply - 4 by all the terms in the first term

x^2 * x - 2x^2 - 3x - 4x^2 + 8x+12

Combine like terms

x^3 - 6x^2 + 5x+12

This is now in standard form, since the exponents go from highest to lowest power

y = x^3 - 6x^2 + 5x+12