Question

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Find all values of k such that f(x) is divisible by the given linear polynomial.

f(x) = kx^3 + x^2 + k^2x + 3k^2 + 3; x + 2

k = ___________

Answer 1

k = 1 or 7

Explanation:

Factor Theorem

If f(x) is a polynomial, and f(a) = 0, then (x – a) is a factor of f(x).

Given polynomial:

`f(x) = kx^3 + x^2 + k^2x + 3k^2 + 3`

According to the factor theorem, if f(x) is divisible by (x + 2) then f(-2) = 0.

Substitute x = -2 into the function and set it to zero:

`\begin{aligned}k(-2)^3+(-2)^2+k^2(-2)+3k^2+3&=0\\-8k+4-2k^2+3k^2+3&=0\\k^2-8k+7&=0\end{aligned}`

Factor the quadratic:

`\begin{aligned}k^2-8k+7&=0\\k^2-k-7k+7&=0\\k(k-1)-7(k-1)&=0\\(k-7)(k-1)&=0\end{aligned}`

Apply the zero-product property:

`\implies k-7=0 \implies k=7`

`\implies k-1=0 \implies k=1`

Therefore, the values of k such that f(x) is divisible by the given linear polynomial are:

• k = 1 or k = 7