Question

Factorise
(i) $a^4-b^4$ (ii) $p^4-81$ (iii) $x^4-(y+z{)}^4$ (iv) $x^4-(x-z{)}^4$ (v) $a^4-2a^2{b}^2+b^4$

Answer 1

$i)a^4-b^4=(a^2-b^2\left)\right(a^2+b^2)=(a+b\left)\right(a-b\left)\right(a^2+b^2)=(a+b\left)\right(a^2+b^2\left)\right(a-b\left)ii\right)p^4-81=(p^2-9)(p^2+9)=(p+3)(p-3)(p^2+9)iii)x^4-{(y+z)}^4=(x^2-{(y+z)}^2\left)\right(x^2+{(y+z)}^2)=(x+y+z\left)\right(x-y-z\left)\right(x^2+{(y+z)}^2\left)iv\right)x^4-{(x-z)}^4=(x^2-{(x-z)}^2)(x^2+{(x-z)}^2)=(x+x-z)(x-x+z)(x^2+{(x-z)}^2)=(2x-z)\left(z\right)(x^2+{(x-z)}^2)v)a^4-2a^2{b}^2+b^4=a^4-a^2{b}^2-a^2{b}^2+b^4=a^2(a^2-b^2)-b^2(a^2-b^2)={(a^2-b^2)}^2={(a+b)}^2{(a-b)}^2$