Find the zeros of the polynomial function below. If a zero is a multiple zero, state its multiplicity. $P ( x ) = x ^ { 3 } - 5 x ^ { 2 } - 15 x + 7$
A) $7,$ $- 1 - \sqrt { 2 },$ $- 1 + \sqrt { 2 }$
B) $7,$ $1 - \sqrt { 2 },$ $1 + \sqrt { 2 }$
C) $- 1,$ $7 - \sqrt { 2 },$ $7 + \sqrt { 2 }$
D) $1,$ $7 - \sqrt { 2 },$ $7 + \sqrt { 2 }$
E) $- 1,$ $- 7 - \sqrt { 2 },$ $- 7 + \sqrt { 2 }$

Question

Quizplus · Accepted Answer

To find the zeros of the polynomial $P(x) = x^3 - 5x^2 - 15x + 7$, we can either factor the polynomial directly (if possible) or use numerical methods or the Rational Root Theorem to find initial guesses and then refine them. However, without doing the algebra here, we can evaluate the given options. The correct zeros can be verified by substituting them back into the polynomial equation and checking if they satisfy the equation $P(x) = 0$. Among the given options, only the zeros in option A, when substituted into the polynomial, will satisfy the equation $P(x) = 0$. This indicates that these are indeed the zeros of the polynomial. The other options do not satisfy the polynomial equation when their values are substituted in, indicating they are not the correct zeros.

Find the Zeros of the Polynomial Function Below P(x)=x3−5x2−15x+7P ( x ) = x ^ { 3 } - 5 x ^ { 2 } - 15 x + 7P(x)=x3−5x2−15x+7

Find the Zeros of the Polynomial Function Below $P ( x ) = x ^ { 3 } - 5 x ^ { 2 } - 15 x + 7$