Question

The polynomial of degree 4, P(x) has a root of multiplicity 2 at x=2 and roots of multiplicity 1 at x=0 and x=−3. Its lead coefficient is 4.
Find a formula for P(x).

Answer 1

Would you agree that if x = 2 then x-2 is a factor?

Answer 2

yes

Answer 3

So, if the root 2 has multiplicity 2, then x-2 must be a factor twice.

Answer 4

so (x-2)(x-2)(x+4)?

Answer 5

And if x = 0 is a root, then wouldn't x-0 be a factor?

Answer 6

And if x = -3 is a root wouldn't x+3 be a factor?

Answer 7

yea you are right

Answer 8

so what do i do then?
FOIL them?

Answer 9

So write the factors that you know so far.

Answer 10

(x-2)(x-2)(x+3)(x-0)

Answer 11

And that product would be 0, do you agree?

Answer 12

im not sure. how so?

Answer 13

Solve this equation:

(x-6)(x+9)=0

Answer 14

x=-9
x=6

Answer 15

So working backwards, we see that if x = -9 is an answer then x+9 is a factor. Similarly, if x = 6 is an answer then x-6 is a factor and the product has to be 0 based on the 0 factor property

Answer 16

So your equation must be 4(x-2)(-2)(x)(x+3)=0

Answer 17

Whether or not you multiply it out depends on what your instructor told you.