If $5x+3$ divides evenly into $10x^3 + x^2 + 32x + k$, find the value of $k$.

Question

If $5x+3$ divides evenly into $10x^3 + x^2 + 32x + k$, find the value of $k$.

This is a polynomial division question and I'm not sure how to do it.

I keep getting the wrong answer. Does anyone have any tips?

Answer 1

Hint: We get

$$\frac{10x^3+x^2+32x+k}{5x+3}=2x^2-x+7+\frac{k-21}{5x+3}$$

Answer 2

Note that if $5x+3 \mid 10x^3 + x^2 + 32x + k$, then we get that $x = -\frac{3}{5}$ is a root of the second polynomial, as it's already a root of the first one. Thus you need to find the value $k$ s.t.

$$0 = 10 \left(-\frac{3}{5}\right)^2 + \left(-\frac{3}{5}\right)^2 + 32\left(-\frac{3}{5}\right) + k$$

Now just find the actual value of $k$.