Question

Find each of the following products.
(i) $(x-4)(x-4)$
(ii) $(2x-3y)(2x-3y)$
(iii) $(\frac{3}{4}x-\frac{5}{6}y)(\frac{3}{4}x-\frac{5}{6}y)$
(iv) $(x-\frac{3}{x})(x-\frac{3}{x})$
(v) $(\frac{1}{3}{x}^2-9)(\frac{1}{3}{x}^2-9)$
(vi) $(\frac{1}{2}{y}^2-\frac{1}{3}y)(\frac{1}{2}{y}^2-\frac{1}{3}y)$

Answer 1

(i) $(x-4)(x-4)=(x-4{)}^2=(x{)}^2-2\times x\times 4+(4{)}^2=x^2-8x+16$

(ii) $(2x-3y)(2x-3y)=(2x-3y{)}^2=(2x{)}^2-2\times 2x\times 3y+(3y{)}^2=4x^2-12xy+9y^2$

(iii) $(\frac{3}{4}x-\frac{5}{6}y)(\frac{3}{4}x-\frac{5}{6}y)={(\frac{3}{4}x-\frac{5}{6}y)}^2={\left(\frac{3}{4}x\right)}^2-2\times \frac{3}{4}x\times \frac{5}{6}y+{\left(\frac{5}{6}y\right)}^2=\frac{9}{16}{x}^2-\frac{5}{4}xy+\frac{25}{36}{y}^2$

(iv) $(x-\frac{3}{x})(x-\frac{3}{x})=(x{)}^2-2\times x\times \frac{3}{x}+{\left(\frac{3}{x}\right)}^2=x^2-6+\frac{9}{x^2}$

(v) $(\frac{1}{3}{x}^2-9)(\frac{1}{3}{x}^2-9){\left(\frac{1}{3}{x}^2\right)}^2-2\times \frac{1}{3}{x}^2\times 9+(9{)}^2=\frac{1}{9}{x}^4-6x^2+81$

(vi) $(\frac{1}{2}{y}^2-\frac{1}{3}y)(\frac{1}{2}{y}^2-\frac{1}{3}y)={(\frac{1}{2}{y}^2-\frac{1}{3}y)}^2={\left(\frac{1}{2}{y}^2\right)}^2-2\times \frac{1}{2}{y}^2\times \frac{1}{3}y+{\left(\frac{1}{3}y\right)}^2=\frac{1}{4}{y}^4-\frac{1}{3}{y}^3+\frac{1}{9}{y}^2$