Question

Find the factors of the following expressions using suitable identity:
(i) $p^2+6p+9$
(ii) $4z^2-4z+1$
(iii) $a^4-8a^2{b}^2+16b^4-81$
(v) $625-x^2-2xy-y^2$

Answer 1

$\left(i\right)p^2+6p+9$

$\Rightarrow p^2+2\times 3\times p+(3{)}^2$ $\left[\right(a+b{)}^2=a^2+2ab+b^2]$

$\Rightarrow (p+3{)}^2$

(ii) $4z^2-4z+1$

$\Rightarrow (2z{)}^2-2\times 2z+(1{)}^2$ $\left[\right(a-b{)}^2=a^2-2ab+b^2]$

(iii) $a^4-8a^2{b}^2+16b^4-81$

$\Rightarrow (a^2{)}^2-2\times 4a^2{b}^2+(4b^2{)}^2-(a{)}^2$ $\left[\right(a-b{)}^2=a^2-2ab+b^2]$

$\Rightarrow (a^2-4b^2{)}^2-(a{)}^2$ $[\because a^2-b^2=(a+b\left)\right(a-b\left)\right]$

$\Rightarrow (a^2-4b^2+a)(a^2-4b^2-a)$

(iv) $625-x^2-2xy-y^2$

$\Rightarrow (25{)}^2-(x^2+2xy+y^2)$ $[\therefore (x+y{)}^2=x^2+2xy+y^2]$

$\Rightarrow (25{)}^2-(x+y{)}^2$

$=(25+x+y)(25-x-y)$