Find a polynomial function with the given real zeros whose graph contains the given point. Zeros: 0 multiplicity 2, 2 multiplicity 3; degree 5, Point: (-1, 27)

Question

Find a polynomial function with the given real zeros whose graph contains the given point. Zeros: 0 multiplicity 2, 2 multiplicity 3; degree 5, Point: (-1, 27)

Answer 1

The zero of a function is defined as x-#=0 so if 0 and 3 are zeroes of a function then the following is true

x=3 and x=0 therefore x-3=0 and x-0=0 or x=0. We will then use the binomials with its multiplicities to create our function:

y=a(x)(x)(x-3)(x-3)(x-3) and to complete the function given the point we will plug into x and y

27=a(-1)(-1)(-1-3)(-1-3)(-1-3)

27=-64a

-27/64=a

therefore in factored form you will have the following

y=-27/64(x)(x)(x-3)(x-3)(x-3)

y=-27/64(x²)(x-3)³

if this needs to be expanded then you will get

y=-27/64(x²)(x²-6x+9)(x-3)

y=-27/64(x²)(x³-9x²+27x-27)

y=-27/64(x⁵-9x⁴+27x³-27x²)

y=-27/64x⁵+243/64x⁴-729/64x³+729/64x²

^{not sure what was needed and hope you understand this response}