Question

Factorise the following!

☺ Solve it☺

☺ Needed Explanation ☺​

Answer 1

Given :

• 24a³ + 81b³
• y³ + 1/8y³
• a³ + 8/a³
• 1 + q³/125

Solution

Apply identity : a³ + b³ = (a + b)(a² - ab + b²)

→ 24a³ + 81b³

Take 3 as a common

→ 3[8a³ + 27b³]

→ 3[(2a)³ + (3b)³]

→ 3[{2a + 3b}{(2a)² - 2a*3b + (3b)²]

→ 3[{2a + 3b}{4a² - 6ab + 9b²}]

• y³ + 1/8y³

Apply identity : a³ + b³ = (a + b)(a² - ab + b²)

→ (y)³ + (1/2y)³

→ (y + 1/2y)[(y)² - y*(1/2y) + (1/2y)²]

→ (y + 1/2)[y² - 1/2 + 1/4y²]

• a³ + 8/a³

Apply identity : a³ + b³ = (a + b)(a² - ab + b²)

→ (a)³ + (2/a)³

→ (a + 2/a)[(a)² - a*2/a + (2/a)²]

→ (a + 2/a)[a² - 2 + 4/a²]

• 1 + q³/125

Apply identity : a³ + b³ = (a + b)(a² - ab + b²)

→ (1)³ + (q/5)³

→ (1 + q/5)[(1)² - 1*q/5 + (q/5)²]

→ (1 + q/5)[(1 - q/5 + q²/25]

Answer 2

Qᴜᴇsᴛɪᴏɴ ❺ :

➥ 24a³ + 81b³ y² + 1/8³

Sᴏʟᴜᴛɪᴏɴ :

$:\implies$ 3(8a³ + 27b³)

$:\implies$ 3[2(a)³ + (3b)³]

$:\implies$ 3(2a + 3b) {(2a)² + (3b)² - 2a*3b}

$:\implies$ $\underline{ \overline{ \boxed{ \purple{ \: \: \bf{3(2a + 3b)(4 {a}^{2} + 9 {b}^{2} - 6ab \: \:}}}}}$

Qᴜᴇsᴛɪᴏɴ ❻ :

➥ y³ + $\sf\dfrac{1}{8{y}^{3}}$

Sᴏʟᴜᴛɪᴏɴ :

$\sf{: \implies ( {y)}^{3} + \left(\dfrac{1}{2y} \right)^{3} }$

$\sf{: \implies \left(y + \dfrac{1}{2y} \right) \left \{ {(y)}^{2} + \left(\dfrac{1}{2y} \right)^{2} - y * \dfrac{1}{2y} \right \}}$

$:\implies$ $\underline{ \overline{ \boxed{ \purple{ \bf{ \sf{ \left(y + \dfrac{1}{2y} \right) \left( {y}^{2} + \dfrac{1}{ {4y}^{2}} - \dfrac{1}{2} \right)}}}}}}$

Qᴜᴇsᴛɪᴏɴ ❼ :

➥ a³ + $\sf\dfrac{8}{{a}^{3}}$

Sᴏʟᴜᴛɪᴏɴ :

$:\implies$ a³ + $\sf\dfrac{8}{{a}^{3}}$

$:\implies$ $\sf{{(a)}^{3} + \left(\dfrac{2}{a}\right) ^{3}}$

$\sf{ : \implies \left( a + \dfrac{2}{a} \right) \left \{ {(a)}^{2} + \left(\dfrac{2}{a} \right)^{2} - a * \dfrac{2}{a} \right \} }$

$:\implies$ $\underline{ \overline{ \boxed{ \purple{ \bf{ \sf{ \left( a + \dfrac{2}{a} \right) \left \{ {a}^{2} + \dfrac{4}{ {a}^{2} } ^{2} - a \right \} }}}}}}$

Qᴜᴇsᴛɪᴏɴ ❽ :

➥ 1 + $\sf\dfrac{{q}^{3}}{{125}}$

Sᴏʟᴜᴛɪᴏɴ :

$:\implies$ 1 + $\sf\dfrac{{q}^{3}}{{125}}$

$\sf{: \implies {(1)}^{3} + \left(\dfrac{q}{5} \right)^{3}}$

$\sf{: \implies \left(1 + \dfrac{q}{5} \right) \left \{ {(1)}^{2} + \left(\dfrac{q}{5} \right)^{2} - 1* \dfrac{q}{5} \right \}}$

$: \implies\underline{ \overline{ \boxed{ \purple{ \bf{ \left(1 + \dfrac{q}{5} \right) \left( 1 + \frac{ {q}^{2} }{25} - \frac{q}{5} \right)}}}}}$