Question

Question 33
Find the following products:
(i) $(\frac{x}{2}+2y)(\frac{x^2}{4}-xy+4y^2)$

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

Answer 1

(i) $(\frac{x}{2}+2y)(\frac{x^2}{4}-xy+4y^2)=\frac{x}{2}(\frac{x^2}{4}-xy+4y^2)+2y(\frac{x^2}{4}-xy+4y^2)$
$\frac{x}{2}\left(\frac{x^2}{4}\right)+\frac{x}{2}(-xy)+\frac{x}{2}\left(4y^2\right)+2y\left(\frac{x^2}{4}\right)+2y(-xy)+2y\left(4y^2\right)$
$\frac{x^3}{8}-\frac{x^{2}y}{2}+2x{y}^2+\frac{x^{2}y}{2}-2x{y}^2+8y^3$
$=\frac{x^3}{8}+8y^3$

(ii) $(x^2-1)(x^4+x^2+1)=x^2(x^4+x^2+1)-1(x^4+x^2+1)$
$=\left[x^2\right(x^4)+x^2(x^2)+x^2(1)-1(x{)}^4-1\left(x^2\right)-1\left(1\right)]$
$=x^6+x^4+x^2-x^4-x^2-1=x^6-1$