SOLUTION: find a polynomnomial of degree 4 with 1 ask a zereo of multiplicity 2 and -3 and 5 as zeros of multiplicity 1. please show your work.

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Question 1138584: find a polynomnomial of degree 4 with 1 ask a zereo of multiplicity 2 and -3 and 5 as zeros of multiplicity 1. please show your work.
Found 2 solutions by Boreal, MathLover1:
Answer by Boreal(15235) About Me  (Show Source):
You can put this solution on YOUR website!
the factors are (x-1)^2(x+3)(x-5)
multiplicity 2 means squared and the other factors give the zeros wished
(x^2-2x+1)(x^2-2x-15)
x^4-2x^3-15x^2-2x^3+4x^2+30x+x^2-2x-15
x^4-4x^3-10x^2+28x-15
graph%28300%2C300%2C-10%2C10%2C-100%2C100%2Cx%5E4-4x%5E3-10x%5E2%2B28x-15%29

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

a polynomnomial of degree 4+ with

1 as a zero of multiplicity 2->a zero appears twice
so you have x%5B1%5D=1 and x%5B2%5D=1
and -3 and 5 as zeros of multiplicity 1->each appears ones
so you have x%5B3%5D=-3 and x%5B4%5D=5
using zero product rule, we have
f%28x%29=%28x-x%5B1%5D%29%28x-x%5B2%5D%29%28x-x%5B3%5D%29%28x-x%5B4%5D%29...substitute given values
f%28x%29=%28x-1%29%28x-1%29%28x-%28-3%29%29%28x-5%29
f%28x%29=%28x%5E2-2x%2B1%29%28x%2B3%29%28x-5%29
f%28x%29=%28x%5E2-2x%2B1%29%28x%5E2-5x%2B3x-15%29
f%28x%29=%28x%5E2-2x%2B1%29%28x%5E2-2x-15%29
f%28x%29=x%5E4+-+4x%5E3+-+10x%5E2+%2B+28x+-+15