Polynomial Division?

Question

The polynomial function is h(x) = f(x) times g(x). Find g(x) if h(2t^3) = 64t^12 - 24t^9 + 104t^6 - 36t^3 + 12 and f(x) = 4x^2 - 3x +2.

Answer 1

h(2t^3) = 64t^12 - 24t^9 + 104t^6 - 36t^3 + 12

Now take out the factor 2t^3 as many times as you can.

h(2t^3) = 4(2t^3)^4 - 3(2t^3)^3 + 26(2t^3)^2 - 18(2t^3) + 12

Now let x = 2t^3

Therefore, h(x) = 4x^4 - 3x^3 + 26x^2 - 18x + 12

We are given: h(x) = f(x) * g(x)

so, g(x) = h(x) / f(x)

= (4x^4 - 3x^3 + 26x^2 - 18x + 12) / (4x^2 - 3x + 2)

Now use synthetic division.

Thus, g(x) = x^2 + 6