How do I contrict a polynomial function?

Question

Fifth degree, 1 is a zero of multiplicity 4, -2 is the only other zero, leading coefficient is 4.

Answer 1

Do you mean, "How do I CONSTRUCT" a polynomial function?

If so, here is how...

5th degree means there are 5 roots. You are given that four of the roots are x=1 (that is the "multiplicity 4) and one of the roots is x=-2. And knowing the leading coefficient = 4 is very helpful.

Each zero can be turned into a factor. x=1, subtracting 1 from both sides gives x-1 = 0.

This makes (x-1) a factor. The other zero is x = -2. Add 2 to both sides and get x+2=0. So the other factor is (x+2). Now we can construct the polynomial. It looks like:

f(x)=4·(x-1)⁴·(x+2) = Ax⁵+ Bx⁴+Cx³+Dx²+Ex+F

Multiply all of this out: (x-1)⁴ = x⁴-4x³+6x²-4x+1

and (x⁴-4x³+6x²-4x+1)·(x+2)=x⁵-2x⁴-2x³+8x²-7x+2

Now we need to multiply in our leading coefficient of 4.

So 4·(x⁵-2x⁴-2x³+8x²-7x+2) will give the final answer of

f(x)=4x⁵-8x⁴-8x³+32x²-28x+8