Question

Find each of the following products:
(i) (x + 3)(x − 3)
(ii) (2x + 5)(2x − 5)
(iii) (8 + x)(8 − x)
(iv) (7x + 11y)(7x − 11y)
(v) $\left(5x^2+\frac{3}{4}{y}^2\right)\left(5x^2-\frac{3}{4}{y}^2\right)$
(vi) $\left(\frac{4x}{5}-\frac{5y}{3}\right)\left(\frac{4x}{5}+\frac{5y}{3}\right)$
(vii) $\left(x+\frac{1}{x}\right)\left(x-\frac{1}{x}\right)$
(viii) $\left(\frac{1}{x}+\frac{1}{y}\right)\left(\frac{1}{x}-\frac{1}{y}\right)$

Answer 1

(i) We have:

$$\begin{gathered} (x+3)(x-3) \\ =x^2-9[\mathrm{using}(a+b\left)\right(a-b)=a^2-b^2] \end{gathered}$$

(ii) We have:

$$\begin{gathered} (2x+5)(2x-5) \\ =4x^2-25[\mathrm{using}(a+b\left)\right(a-b)=a^2-b^2] \end{gathered}$$

(iii) We have:

$$\begin{gathered} (8+x)(8-x) \\ =64-x^2[\mathrm{using}(a+b\left)\right(a-b)=a^2-b^2] \end{gathered}$$

(iv) We have:

$$\begin{gathered} (7x+11y)(7x-11y) \\ =49x^2-121y^2[\mathrm{using}(a+b\left)\right(a-b)=a^2-b^2] \end{gathered}$$

(v) We have:

$$\begin{gathered} \left(5x^2+\frac{3}{4}{y}^2\right)\left(5x^2-\frac{3}{4}{y}^2\right) \\ =25x^4-\frac{9}{16}{y}^4[\mathrm{using}(a+b\left)\right(a-b)=a^2-b^2] \end{gathered}$$

(vi) We have:

$$\begin{gathered} \left(\frac{4x}{5}-\frac{5y}{3}\right)\left(\frac{4x}{5}+\frac{5y}{3}\right) \\ =\frac{16x^2}{25}-\frac{25y^2}{9}[\mathrm{using}(a+b\left)\right(a-b)=a^2-b^2)] \end{gathered}$$

(vii) We have:
$$\begin{gathered} \left(x+\frac{1}{x}\right)\left(x-\frac{1}{x}\right) \\ =x^2-\frac{1}{x^2}[\mathrm{using}(a+b\left)\right(a-b)=a^2-b^2] \end{gathered}$$

(viii) We have:
$$\begin{gathered} \left(\frac{1}{x}+\frac{1}{y}\right)\left(\frac{1}{x}-\frac{1}{y}\right) \\ =\frac{1}{x^2}-\frac{1}{y^2}[\mathrm{using}(a+b\left)\right(a-b)=a^2-b^2] \end{gathered}$$

(ix) We have:
$$\begin{gathered} \left(2a+\frac{3}{b}\right)\left(2a-\frac{3}{b}\right) \\ =4a^2-\frac{9}{b^2}[\mathrm{using}(a+b\left)\right(a-b)=a^2-b^2] \end{gathered}$$