Question

Simplify:

(i) $3c(2b-2a)+2a(b+3c)-2b(3c+a)$

(ii) $m\left(m^3-m^2+1\right)-2\left(m^4-m^3-1\right)+m^2\left(m^2-m+2\right)$

(iii) $2\left(x^2+y-3\right)+7\left(x^2-y+2\right)-3\left(y-3x^2-1\right)$

(iv) $\left(x+y\right)\left(x-y\right)+\left(2x-y\right)\left(3x+y\right)$

(v) $\left(2a+b\right)\left(c-2d\right)+\left(a-b\right)\left(2c+3d\right)+4\left(ac+bd\right)$

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right) \\ \mathrm{We}\mathrm{have}3c\left(2b-2a\right)+2a\left(b+3c\right)-2b\left(3c+a\right) \\ =6bc-6ac+2ab+6ac-6bc-2ab \\ =6bc-6bc-6ac+6ac+2ab-2ab \\ =0 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right) \\ \mathrm{We}\mathrm{have}m\left(m^3-m^2+1\right)-2\left(m^4-m^3-1\right)+m^2\left(m^2-m+2\right) \\ =m^4-m^3+m-2m^4+2m^3+2+m^4-m^3+2m^2 \\ =m^4-2m^4+m^4-m^3+2m^3-m^3+2m^2+m+2 \\ =2m^2+m+2 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iii}\right) \\ \mathrm{We}\mathrm{have}2\left(x^2+y-3\right)+7\left(x^2-y+2\right)-3\left(y-3x^2-1\right) \\ =2x^2+2y-6+7x^2-7y+14-3y+9x^2+3 \\ =2x^2+7x^2+9x^2+2y-7y-3y-6+14+3 \\ =18x^2-8y+11 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iv}\right) \\ \mathrm{We}\mathrm{have}\left(x+y\right)\left(x-y\right)+\left(2x-y\right)\left(3x+y\right) \\ =x\left(x-y\right)+y\left(x-y\right)+2x\left(3x+y\right)-y\left(3x+y\right) \\ =x^2-xy+xy-y^2+6x^2+2xy-3xy-y^2 \\ =x^2+6x^2-xy+xy+2xy-3xy-y^2-y^2 \\ =7x^2-xy-2y^2 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{v}\right) \\ \mathrm{We}\mathrm{have}\left(2a+b\right)\left(c-2d\right)+\left(a-b\right)\left(2c+3d\right)+4\left(ac+bd\right) \\ =2a\left(c-2d\right)+b\left(c-2d\right)+a\left(2c+3d\right)-b\left(2c+3d\right)+4\left(ac+bd\right) \\ =2ac-4ad+bc-2bd+2ac+3ad-2bc-3bd+4ac+4bd \\ =2ac+2ac+4ac-4ad+3ad+bc-2bc-2bd-3bd+4bd \\ =8ac-ad-bc-bd \end{gathered}$$