Question

The polynomial $f\left(x\right)=3x^4+5x^3-17x^2-25x+10$ has irrational zeros at $+\sqrt{5}$ and $-\sqrt{5}$. Find the other zeros.

• A. $-2$
• B. $2$
• C. $\frac{1}{3}$
• D. $\frac{2}{3}$

Answer 1

Answer: (A), (C)

$\frac{1}{3}$
$-2$

$f\left(x\right)=3x^4+5x^3-17x^2-25x+10$

Irrational roots are $-\sqrt{5}$ and $\sqrt{5}$

Since $\pm \sqrt{5}$ are roots,

$(x-\sqrt{5})(x+\sqrt{5})$ is a factor of $f\left(x\right)$

$=x^2$

$f\left(x\right)=3x^4-15x^2+5x^3-25x-2x^2+10$

$=3x^2(x^2-5)+5x(x^2-5)-2(x^2-5)$

$=(x^2-5)(3x^2+5x-2)$

So, $3x^2+5x-2$ gives us the remaining roots.

$=3x^2+6x-x-2$

$=3x(x+2)-1(x+2)$

$x=\frac{1}{3},-2$