Find all of the real zeros of the function. Give exact values whenever possible. Identify each zero as rational or irrational.
-$$f ( x ) = x ^ { 3 } - 7 x ^ { 2 } + 10 x + 6$$
A) 3 (rational), $\sqrt { 6 }$ (irrational), and $- \sqrt { 6 }$ (irrational)
B) 3 (rational), $1 + \sqrt { 6 }$ (irrational), and $1 - \sqrt { 6 }$ (irrational)
C) 3 (rational), $- 2 + \sqrt { 6 }$ (irrational), and $- 2 - \sqrt { 6 }$ (irrational)
D) 3 (rational), $2 + \sqrt { 6 }$ (irrational), and $2 - \sqrt { 6 }$ (irrational)

Question

Quizplus · Accepted Answer

The polynomial can be factored as $(x - 3)(x^2 - 4x - 2)$. Solving $x^2 - 4x - 2 = 0$ using the quadratic formula gives the roots $2 \pm \sqrt{6}$, which are irrational. The root $x = 3$ is rational.

Find All of the Real Zeros of the Function f(x)=x3−7x2+10x+6f ( x ) = x ^ { 3 } - 7 x ^ { 2 } + 10 x + 6f(x)=x3−7x2+10x+6

Find All of the Real Zeros of the Function $f ( x ) = x ^ { 3 } - 7 x ^ { 2 } + 10 x + 6$