Question 2 Find all zeros of the function f(x)=8x^3-14x^2-115x-75 Enter the zeros separated by commas. square

To find the zeros of the function $f(x)=8x^{3}-14x^{2}-115x-75$, we need to solve the equation $f(x)=0$.<br /><br />Step 1: Set the function equal to zero:<br />$8x^{3}-14x^{2}-115x-75=0$<br /><br />Step 2: Use the Rational Root Theorem to find possible rational zeros. The Rational Root Theorem states that any rational zero of a polynomial function will have a numerator that is a factor of the constant term and a denominator that is a factor of the leading coefficient.<br /><br />In this case, the constant term is -75 and the leading coefficient is 8. The factors of -75 are $\pm1, \pm3, \pm5, \pm15, \pm25, \pm75$ and the factors of 8 are $\pm1, \pm2, \pm4, \pm8$. Therefore, the possible rational zeros are $\pm1, \pm3, \pm5, \pm15, \pm25, \pm75, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{5}{2}, \pm\frac{15}{2}, \pm\frac{25}{2}, \pm\frac{75}{2}, \pm\frac{1}{4}, \pm\frac{3}{4}, \pm\frac{5}{4}, \pm\frac{15}{4}, \pm\frac{25}{4}, \pm\frac{75}{4}, \pm\frac{1}{8}, \pm\frac{3}{8}, \pm\frac{5}{8}, \pm\frac{15}{8}, \pm\frac{25}{8}, \pm\frac{75}{8}$.<br /><br />Step 3: Test the possible rational zeros by substituting them into the equation $8x^{3}-14x^{2}-115x-75=0$.<br /><br />After testing, we find that the zeros of the function are $x=-3, x=-\frac{5}{4}, x=\frac{5}{2}$.<br /><br />Therefore, the zeros of the function $f(x)=8x^{3}-14x^{2}-115x-75$ are $-3, -\frac{5}{4}, \frac{5}{2}$.