Question

Q)
Factorise the below given algebraic expression both by using identity and also by grouping the common factors.
r²+p²+4q²-4pq-4qr+2rp

Answer 1

$$\begin{gathered} \mathbf{BY}\mathbf{}\mathbf{USING}\mathbf{}\mathbf{IDENTITY}\mathbf{}\mathbf{:} \\ {\mathrm{r}}^2+{\mathrm{p}}^2+4{\mathrm{q}}^2-4\mathrm{pq}-4\mathrm{qr}+2\mathrm{rp} \\ ={\mathrm{p}}^2+4{\mathrm{q}}^2+{\mathrm{r}}^2-4\mathrm{pq}-4\mathrm{qr}+2\mathrm{rp} \\ ={\left(\mathrm{p}\right)}^2+{\left(-2\mathrm{q}\right)}^2+{\left(\mathrm{r}\right)}^2+2\left(\mathrm{p}\right)\left(-2\mathrm{q}\right)+2\left(-2\mathrm{q}\right)\left(\mathrm{r}\right)+2\left(\mathrm{r}\right)\left(\mathrm{p}\right) \\ ={\left(\mathrm{p}-2\mathrm{q}+\mathrm{r}\right)}^2 \\ =\left(\mathrm{p}-2\mathrm{q}+\mathrm{r}\right)\left(\mathrm{p}-2\mathrm{q}+\mathrm{r}\right)\left[{\mathbf{a}}^{\mathbf{2}}\mathbf{+}{\mathbf{b}}^{\mathbf{2}}\mathbf{+}{\mathbf{c}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathbf{ab}\mathbf{+}\mathbf{2}\mathbf{bc}\mathbf{+}\mathbf{2}\mathbf{ca}\mathbf{}\mathbf{=}\mathbf{}{\left(\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{+}\mathbf{c}\right)}^{\mathbf{2}}\right] \\ \mathbf{BY}\mathbf{}\mathbf{GROUPING}\mathbf{}\mathbf{:} \\ {\mathrm{r}}^2+{\mathrm{p}}^2+4{\mathrm{q}}^2-4\mathrm{pq}-4\mathrm{qr}+2\mathrm{rp} \\ ={\mathrm{p}}^2-2\mathrm{pq}+\mathrm{pr}-2\mathrm{pq}+4{\mathrm{q}}^2-2\mathrm{qr}+\mathrm{pr}-2\mathrm{qr}+{\mathrm{r}}^2 \\ =\left({\mathrm{p}}^2-2\mathrm{pq}+\mathrm{pr}\right)-\left(2\mathrm{pq}-4{\mathrm{q}}^2+2\mathrm{qr}\right)+\left(\mathrm{pr}-2\mathrm{qr}+{\mathrm{r}}^2\right) \\ =\mathrm{p}\left(\mathrm{p}-2\mathrm{q}+\mathrm{r}\right)-2\mathrm{q}\left(\mathrm{p}-2\mathrm{q}+\mathrm{r}\right)+\mathrm{r}\left(\mathrm{p}-2\mathrm{q}+\mathrm{r}\right) \\ =\left(\mathrm{p}-2\mathrm{q}+\mathrm{r}\right)\left(\mathrm{p}-2\mathrm{q}+\mathrm{r}\right) \end{gathered}$$