Find all y-intercepts and x-intercepts of the graph of the function. f(x)=2x^3+2x^2-24x If there is more than one answer, separate them with commas. Click on "None" if applicable. y -intercept(s): square x -intercept(s): square

To find the y-intercept of the function $f(x)=2x^{3}+2x^{2}-24x$, we need to evaluate the function at $x=0$.<br /><br />Substituting $x=0$ into the function, we get:<br />$f(0)=2(0)^{3}+2(0)^{2}-24(0)=0$<br /><br />Therefore, the y-intercept is $(0,0)$.<br /><br />To find the x-intercepts of the function $f(x)=2x^{3}+2x^{2}-24x$, we need to solve the equation $f(x)=0$.<br /><br />Setting the function equal to zero, we have:<br />$2x^{3}+2x^{2}-24x=0$<br /><br />Factoring out the common factor of $2x$, we get:<br />$2x(x^{2}+x-12)=0$<br /><br />Now, we can set each factor equal to zero and solve for $x$:<br />$2x=0$ or $x^{2}+x-12=0$<br /><br />Solving the first equation, we get:<br />$x=0$<br /><br />Solving the quadratic equation $x^{2}+x-12=0$, we can use the quadratic formula:<br />$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$<br /><br />Plugging in the values $a=1$, $b=1$, and $c=-12$, we get:<br />$x=\frac{-1\pm\sqrt{1^{2}-4(1)(-12)}}{2(1)}$<br />$x=\frac{-1\pm\sqrt{1+48}}{2}$<br />$x=\frac{-1\pm\sqrt{49}}{2}$<br />$x=\frac{-1\pm7}{2}$<br /><br />Therefore, the x-intercepts are $x=0$, $x=\frac{-1+7}{2}=\frac{6}{2}=3$, and $x=\frac{-1-7}{2}=\frac{-8}{2}=-4$.<br /><br />So, the y-intercept is $(0,0)$ and the x-intercepts are $0$, $3$, and $-4$.