Question

Find the following products:

(i) (3x + 2y) (9x² − 6xy + 4y²)

(ii) (4x − 5y) (16x² + 20xy + 25y²)

(iii) (7p⁴ + q) (49p⁸ − 7p⁴q + q²)

(iv) $\left(\frac{x}{2}+2y\right)\left(\frac{x^2}{4}-xy+4y^2\right)$

(v) $\left(\frac{3}{x}-\frac{5}{y}\right)\left(\frac{9}{x^2}+\frac{25}{y^2}+\frac{15}{xy}\right)$

(vi) $\left(3+\frac{5}{x}\right)\left(9-\frac{15}{x}+\frac{25}{x^2}\right)$

(vii) $\left(\frac{2}{x}+3x\right)\left(\frac{4}{x^2}+9x^2-6\right)$

(viii) $\left(\frac{3}{x}-2x^2\right)\left(\frac{9}{x^2}+4x^4-6x\right)$

(ix) (1 − x) (1+ x + x²)

(x) (1 + x) (1 − x + x²)

(xi) (x² − 1) (x⁴ + x² + 1)

(xii) (x³ + 1) (x⁶ − x^{3 +} 1)

Answer 1

(i) In the given problem, we have to find the value of

Given

We shall use the identity

We can rearrange the as

Hence the Product value of is

(ii) Given

We shall use the identity

We can rearrange the as

Hence the Product value of is

(iii) Given

We shall use the identity

We can rearrange the as

Hence the Product value of is

(iv) Given

We shall use the identity

We can rearrange the as

Hence the Product value of is

(v) Given

We shall use the identity

We can rearrange the as

$$\begin{gathered} =\left(\frac{3}{x}\right)\times \left(\frac{3}{x}\right)\times \left(\frac{3}{x}\right)-\left(\frac{5}{y}\right)\times \left(\frac{5}{y}\right)\times \left(\frac{5}{y}\right) \\ =\frac{27}{x^3}-\frac{125}{y^3} \end{gathered}$$

Hence the Product value of is

(vi) Given

We shall use the identity ,

we can rearrange the as

Hence the Product value of is

(vii) Given

We shall use the identity,

We can rearrange the as

Hence the Product value of is

(viii) Given

We shall use the identity

We can rearrange the as

$$\begin{gathered} \left(\frac{3}{x}-2x^2\right)\left({\left(\frac{3}{x}\right)}^2+{\left(2x^2\right)}^2-\left(\frac{3}{x}\right)\left(2x^2\right)\right) \\ ={\left(\frac{3}{x}\right)}^3-{\left(2x^2\right)}^3 \\ =\left(\frac{3}{x}\right)\left(\frac{3}{x}\right)\left(\frac{3}{x}\right)-\left(2x^2\right)\left(2x^2\right)\left(2x^2\right) \\ =\frac{27}{x^3}-8x^6 \end{gathered}$$

Hence the Product value of is

(ix) Given

We shall use the identity

We can rearrange the as

Hence the Product value of is

(x) Given

We shall use the identity

We can rearrange the as

Hence the Product value of is

(xi) Given

We shall use the identity

We can rearrange the as

$\left(x^2-1\right)\left[{\left(x^2\right)}^2+\left(x^2\right)\left(1\right)+{\left(1\right)}^2\right]$

Hence the Product value of is

(xii) Given

We shall use the identity,

We can rearrange the as

Hence the Product value of is .