Find a rational zero of the polynomial function and use it to find all the zeros of the function.
-$$f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 5 x - 6$$
A) $\{ - 3 , - 1,2 \}$
B) $\{ - 2,1,3 \}$
C) $\{ - 3 \}$
D) $\{ - 1 \}$

Question

Quizplus · Accepted Answer

To find a rational zero, we can use the Rational Root Theorem, which suggests that any rational root, in the form of p/q (where p is a factor of the constant term and q is a factor of the leading coefficient), should be tested. For the given polynomial, the constant term is -6, and the leading coefficient is 1. Thus, the possible rational roots are factors of -6, which are ±1, ±2, ±3, and ±6. Testing these values, we find that $x = -3$ is a root because $f(-3) = (-3)^3 + 2(-3)^2 - 5(-3) - 6 = 0$. Using synthetic division or polynomial division with this root, we can factor the polynomial as $(x + 3)(x^2 - x - 2)$. Further factoring or using the quadratic formula on $x^2 - x - 2$ gives the other two roots, $x = -1$ and $x = 2$. Therefore, the complete set of zeros is $\{-3, -1, 2\}$.

Find a Rational Zero of the Polynomial Function and Use f(x)=x3+2x2−5x−6f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 5 x - 6f(x)=x3+2x2−5x−6

Find a Rational Zero of the Polynomial Function and Use $f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 5 x - 6$