Question

Simplify each of the following products:
i) ${(m+\frac{n}{7})}^3(m-\frac{n}{7})$
ii) $(\frac{x}{2}-\frac{2}{5})(\frac{2}{5}-\frac{x}{2})-x^2+2x$

Answer 1

i) ${(m+\frac{n}{7})}^3(m-\frac{n}{7})={(m+\frac{n}{7})}^2(m+\frac{n}{7})(m-\frac{n}{7})={(m+\frac{n}{7})}^2(m^2-\frac{n^2}{49})=(m^2+\frac{n^2}{49}+\frac{2mn}{7})(m^2-\frac{n^2}{49})=m^4+\text{/}\frac{m^2{n}^2}{49}+\frac{2m^{3}n}{7}-\text{/}\frac{m^2{n}^2}{49}-\frac{n^4}{(49{)}^2}-\frac{2m{n}^3}{7\times 49}=m^4-\frac{n^4}{(49{)}^2}+\frac{2mn}{7}(m^2-\frac{n^2}{49})=(m^2-\frac{n^2}{49})(m^2+\frac{n^2}{49}+\frac{2mn}{7})$

ii) $(\frac{x}{2}-\frac{2}{5})(\frac{2}{5}-\frac{x}{2})-x^2+2x=-(\frac{x}{2}-\frac{3}{5})(\frac{x}{2}-\frac{2}{5})=x^2+2x=-\left\{(\frac{x}{2}-\frac{2}{5})\right\}=x^2+2xor{x}^2+2x+{(\frac{x}{2}-\frac{2}{5})}^2=0x^2+2x+(\frac{x^2}{4}+\frac{4}{25}-\frac{2x}{5})\frac{5x^2}{4}+\frac{8x}{5}+\frac{4}{25}=0\Rightarrow \frac{125x^2+160x+16}{100}=0\Rightarrow 125x^2+160x+16=0$