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Suppose $5x - 2$ is a factor of $x^3 - 3x^2 + kx + 15$. Find $k$.
I've tried getting the $x$ value of the factor $5x - 2 = 0$ and got $x= \frac25$ and replaced all the other $x$s with $\frac25$ and set the equation to be equal to $0$. But I don't know if I'm doing right, show me the steps and value of $x$.
Your description of your work makes it sound like you're on the right track.
Let $f(x)=x^3-3x^2+kx+15$. If $5x-2$ is a factor of $f(x)$, then by the remainder theorem, we have $f(\frac25)=0$. Thus,
\begin{align} f(x)&=x^3-3x^2+kx+15\\ f(\frac25)=0&=(\frac25)^3-3(\frac25)^2+k(\frac25)+15\\ 0\color{blue}{\cdot125}&=\bigg(\frac8{125}-\frac{12}{25}+\frac25k+15\bigg)\color{blue}{\cdot125}\\ 0&=8-60+50k+1875\\ -1823&=50k\\ k&=-\frac{1823}{50} \end{align}
The moral of the story: don't be alarmed if your final answer is not a number you expected! Rational numbers are real numbers, too!
Yes the method is fine and we obtain
$$\frac8{125} - 3\frac{4}{25} + k\frac25 + 15=0 \iff8-60+50k+1875 \iff k=-\frac{1823}{50}$$
and indeed
$$x^3 - 3x^2 -\frac{1823}{50}x + 15=\frac1{50} (5 x - 2) (10 x^2 - 26 x - 375) $$
