Find a rational zero of the polynomial function and use it to find all the zeros of the function.
-$$f ( x ) = x ^ { 3 } + 2 x ^ { 2 } + 5 x - 26$$
A) $\{ 2 , - 2 + 3 i , - 2 - 3 i \}$
B) $\{ 2,3 + 2 i , 3 - 2 i \}$
C) $\{ - 2 , - 2 + \sqrt { 5 } , - 4 - \sqrt { 5 } \}$
D) $\{ 2,3 + \sqrt { 5 } , 3 - \sqrt { 5 } \}$

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 a factor of 26. Testing possible rational roots, we find that $x = 2$ is a root because $f(2) = 2^3 + 2(2)^2 + 5(2) - 26 = 0$. After dividing the polynomial by $x - 2$, we get a quadratic equation, which when solved gives the other two roots as complex numbers $-2 + 3i$ and $-2 - 3i$.

Find a Rational Zero of the Polynomial Function and Use f(x)=x3+2x2+5x−26f ( x ) = x ^ { 3 } + 2 x ^ { 2 } + 5 x - 26f(x)=x3+2x2+5x−26

Find a Rational Zero of the Polynomial Function and Use $f ( x ) = x ^ { 3 } + 2 x ^ { 2 } + 5 x - 26$