Question

write a polynomial f(x) of lowest degree that satisfies the given conditions. Zeros: -4, multiplicity 1; 3 multiplicity 2; and f(0)=-108

Answer 1

First start with the factors

(3x+5)(3x+5)(6x-1) The multiplicity of 2 means there are 2 of that same factor.

Next, figure out how to get it to go through the point (0,50) Plug in these points, solve for a

50=a(3(0)+5)(3(0)+5)(6(0)-1)

50=a(5)(5)(-1)

50=-25a Divide both sides by -25

-2=a Plug this in

y=-2(3x+5)(3x+5)(6x-1) Now multiply the factors out. Will start by FOILING the first 2 factors

y=-2(9x2+30x+25)(6x-1) Now multiply the remaining factors together

y=-2(54x3+180x2+150x-9x2-30x-25) Combine like terms

y=-2(54x3+171x2+120x-25) Distribute the -2

y=-108x3-342x2-240x+50