Question

Find the product of the following binomials:
(i) (2x + y)(2x + y)
(ii) (a + 2b)(a − 2b)
(iii) (a² + bc)(a² − bc)
(iv) $\left(\frac{4x}{5}-\frac{3y}{4}\right)\left(\frac{4x}{5}+\frac{3y}{4}\right)$
(v) $\left(2x+\frac{3}{y}\right)\left(2x-\frac{3}{y}\right)$
(vi) (2a³ + b³)(2a³ − b³)
(vii) $\left(x^4+\frac{2}{x^2}\right)\left(x^4-\frac{2}{x^2}\right)$
(viii) $\left(x^3+\frac{1}{x^3}\right)\left(x^3-\frac{1}{x^3}\right)$

Answer 1

(i) We will use the identity ${\left(a+b\right)}^2=a^2+2ab+b^2$ in the given expression to find the product.
$$\begin{gathered} \left(2x+y\right)\left(2x+y\right) \\ ={\left(2x+y\right)}^2 \\ ={\left(2x\right)}^2+2\left(2x\right)\left(y\right)+y^2 \\ =4x^2+4xy+y^2 \end{gathered}$$

(ii) We will use the identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$ in the given expression to find the product.
$$\begin{gathered} \left(a+2b\right)\left(a-2b\right) \\ =a^2-{\left(2b\right)}^2 \\ =a^2-4b^2 \end{gathered}$$

(iii) We will use the identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$ in the given expression to find the product.
$$\begin{gathered} \left(a^2+bc\right)\left(a^2-bc\right) \\ ={\left(a^2\right)}^2-{\left(bc\right)}^2 \\ =a^4-b^2{c}^2 \end{gathered}$$

(iv)We will use the identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$ in the given expression to find the product.
$$\begin{gathered} \left(\frac{4x}{5}-\frac{3y}{4}\right)\left(\frac{4x}{5}+\frac{3y}{4}\right) \\ ={\left(\frac{4x}{5}\right)}^2-{\left(\frac{3y}{4}\right)}^2 \\ =\frac{16x^2}{25}-\frac{9y^2}{16} \end{gathered}$$

(v) We will use the identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$ in the given expression to find the product.
$$\begin{gathered} \left(2x+\frac{3}{y}\right)\left(2x-\frac{3}{y}\right) \\ ={\left(2x\right)}^2-{\left(\frac{3}{y}\right)}^2 \\ =4x^2-\frac{9}{y^2} \end{gathered}$$

(vi) We will use the identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$ in the given expression to find the product.
$$\begin{gathered} \left(2a^3+b^3\right)\left(2a^3-b^3\right) \\ ={\left(2a^3\right)}^2-{\left(b^3\right)}^2 \\ =4a^6-b^6 \end{gathered}$$

(vii) We will use the identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$ in the given expression to find the product.
$$\begin{gathered} \left(x^4+\frac{2}{x^2}\right)\left(x^4-\frac{2}{x^2}\right) \\ ={\left(x^4\right)}^2-{\left(\frac{2}{x^2}\right)}^2 \\ =x^8-\frac{4}{x^4} \end{gathered}$$

(viii) We will use the identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$ in the given expression to find the product.

$$\begin{gathered} \left(x^3+\frac{1}{x^3}\right)\left(x^3-\frac{1}{x^3}\right) \\ ={\left(x^3\right)}^2-{\left(\frac{1}{x^3}\right)}^2 \\ =x^6-\frac{1}{x^6} \end{gathered}$$