Question

Find the square root of the polynomial $4+25x^2-12x-24x^3+16x^4$ by division method.

Answer 1

$4+25x^2-12x-24x^3+16x^4$ written in order as

$16x^4-24x^3+25x^2-12x+4$

If this is a square of a simpler polynomial, then it must be expressible in one of the two forms.

${(4x^2+ax-2)}^2$ or ${(4x^2+ax+2)}^2$

So we have one of the following:

Case$1:{(4x^2+ax-2)}^2=16x^4+8a{x}^3+(a^2-8)x^2-4ax+4$ on expanding

Again Case$2:{(4x^2+ax+2)}^2=16x^4+8a{x}^3+(a^2+8)x^2+4ax+4$ on expanding

Equating the coefficeints in each case

$16x^4-24x^3+25x^2-12x+4=16x^4+8a{x}^3+(a^2\pm 8)x^2\pm 4ax+4$

we get $8a=-24\Rightarrow a=\frac{-24}{8}=-3$

and $\pm 4a=-12\Rightarrow a=\pm 3$

So,$16x^4-24x^3+25x^2-12x+4={(4x^2-3x+2)}^2$

$\therefore \sqrt{16x^4-24x^3+25x^2-12x+4}=4x^2-3x+2$