Question

A quadratic polynomial $p\left(x\right)$ has $1+\sqrt{5}$ and $1-\sqrt{5}$ as its zeros and $p\left(1\right)=2$. Then the value of $p\left(0\right)$ is

• A. $-\frac{4}{5}$
• B. $\frac{8}{5}$
• C. $-\frac{8}{5}$
• D. $\frac{4}{5}$

Answer 1

The correct option is B $\frac{8}{5}$

Let
$p\left(x\right)=a(x-(1+\sqrt{5}\left)\right)(x-(1-\sqrt{5}\left)\right)$
$p\left(x\right)=a(x^2-x(1+\sqrt{5}+1-\sqrt{5})+(1-5\left)\right)p\left(x\right)=a(x^2-2x-4)$
Now,
$p\left(1\right)=2$
$\Rightarrow 2=a(1-2-4)\Rightarrow a=-\frac{2}{5}$
$\Rightarrow p\left(x\right)=-\frac{2}{5}(x^2-2x-4)\therefore p\left(0\right)=\frac{8}{5}$