Question

question 7 of 25
which of the following are roots of the polynomial function below?
check all that apply.
$f(x) = 2x^3 - 5x^2 + 2x + 1$

a. $\frac{3 - sqrt{17}}{4}$
b. $1$
c. $\frac{5 + sqrt{10}}{6}$
d. $\frac{5 - sqrt{10}}{6}$
e. $\frac{3 + sqrt{17}}{4}$

Explanation:

Step1: Test integer root candidate 1

Substitute $x=1$ into $F(x)$:
$F(1)=2(1)^3 -5(1)^2 +2(1)+1=2-5+2+1=0$. So $x=1$ is a root.

Step2: Factor polynomial using root 1

Divide $F(x)$ by $(x-1)$:
$2x^3 -5x^2 +2x +1=(x-1)(2x^2 -3x -1)$.

Step3: Solve quadratic $2x^2 -3x -1=0$

Use quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ where $a=2,b=-3,c=-1$:
$x=\frac{3\pm\sqrt{9+8}}{4}=\frac{3\pm\sqrt{17}}{4}$.

Answer:

A. $\frac{3-\sqrt{17}}{4}$, B. 1, E. $\frac{3+\sqrt{17}}{4}$