Question

$2$ and $-2$ are two zeros of the polynomial $m^4+m^3-34m^2-4m+120$. What are the other two zeros of the polynomial?

• A. $5,-6$
• B. $-5,6$
• C. $-5,-6$
• D. $5,6$

Answer 1

The correct option is A $5,-6$

$m^4+m^3-34m^2-4m+120=f\left(m\right)$
$2$ & $-2$ are two zeroes of polynomial
Let $\alpha ,\beta$ be other two zeroes
Sum of roots$=\alpha +\beta +2-2=-1=\frac{-Coeff.of{m}^3}{Coeff.of{m}^4}$
$\alpha +\beta =-1⟶1$
Product of roots$=\alpha \beta \left(2\right)(-2)=\frac{constantterm}{Coeff.of{m}^4}$
$-4\alpha \beta =120$
$\alpha \beta =-30⟶2$
${(\alpha -\beta )}^2={(\alpha +\beta )}^2-4\alpha \beta$
${(\alpha -\beta )}^2={(-1)}^2-4(-30)$
${(\alpha -\beta )}^2=1+120$
$\alpha -\beta =\pm 11$
$I.\alpha -\beta =11⟶3$
Add 1 & 3
$2\alpha =10$
$\alpha =5$
$\therefore \beta =-6$
$II.\alpha -\beta =-11⟶4$
Add 1 & 4
$2\alpha =-12$
$\alpha =-6$
$\beta =5$
$\therefore 5$ and $-6$ are other two zeroes.