Question

For what value of k, is the polynomial f(x) = 3x⁴ – 9x³ + x² + 15x + k completely divisible by 3x² – 5?

Answer 1

Given: f(x) = 3x⁴ – 9x³ + x² + 15x + k is divisible by 3x² – 5.

Step-wise calculation:

1. If 3x² – 5 divides f(x), then every root of 3x² – 5 = 0 `("so"  x^2 = 5/3)` is a root of f(x).

2. For `x^2 = 5/3` we have `x^4 = (5/3)^2 = 25/9`.

Evaluate f: `f = 3(25/9) - 9x^3 + 5/3 + 15x + k`

= `25/3 + 5/3 - 9x^3 + 15x + k`

= 10 – 9x³ + 15x + k

3. But `x^3 = x xx x^2 = x xx 5/3`.

So `-9x^3 = -9 xx (5/3)x` = –15x, which cancels the +15x term. 

Thus f = 10 + k.

4. For f to be zero at the roots we need 10 + k = 0 ⇒ k = –10.