Question

if m and n are zeros of 4x²+4x+1 then form a quadratic polynomial whose zeros are 2m and 2n



quick ​

Answer 1

Explanation:

$4x^2+4x+1$

$=4x^2+2x+2x+1$

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

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

∴ $m = n = \frac{-1}{2}$

In a quadratic polynomial of form $ax^2+bx+c$ where $\alpha$ and $\beta$ are its roots,

$\alpha +\beta =\frac{-b}{a}$

$\alpha \beta =\frac{c}{a}$

$2m = 2n = 2\times\frac{-1}{2}=-1$

Hence, if we consider a to be 1,

$2m+2n=-1-1=-b$

==> $-b=-2$

∴ $b=2$

$c=2m\times2n=-1\times-1$

∴ $c=1$

Hence, the quadratic polynomial so formed is $x^2 +2x + 1$.

Answer: x²+2x+1