Question

$\sf{Question}$
Fatorises the given expression
1) x³ + 3x² + 3x + 28
2) x² + 14x + 48
3) x⁴ + 16​

Answer 1

Given :-

• x³ + 3x² + 3x + 28

Explaination :-

⇒ x³ + 3x² + 3x + 1 + 27

• (a + b)³ = a³ + b³ + 3a²b + 3ab²

⇒ (x + 1)³ + (3)³

• a³ + b³ = (a + b){a² - ab + b²}

⇒ (x + 1 + 3){(x + 1)² - (x + 1)3 + (3)²}

⇒ (x + 4){x² + 2x + 1 - 3x - 3 + 9}

⇒ (x + 4)(x² - x + 7) Ans.

Given :-

• x² + 14x + 48

Explaination :-

⇒ x² + 14x + 48

• Splitting middle term

⇒ x² + 8x + 6x + 48

⇒ x(x + 8) + 6(x + 8)

⇒ (x + 6)(x + 8) Ans.

Given :-

• x⁴ + 16

Explaination :-

⇒ x⁴ + 16

⇒ (x²)² + (4)²

• (a + b)² = a² + b² + 2ab
• a² + b² = (a + b)² - 2ab

⇒ (x²)² + (4)² + 2 × x² × 4 + 4² - 8x²

⇒ (x² + 4)² - 8x²

⇒ (x² + 4)² - (2√2x)²

• a² - b² = (a + b)(a - b)

⇒ (x² + 4 + 2√2x)(x² + 4 - 2√2x)

⇒ (x² + 2√2x + 4)(x² - 2√2x + 4) Ans.

Answer 2

Given :-

1) x³ + 3x² + 3x + 28

2) x² + 14x + 48

3) x⁴ + 16

To Do :-

Factorization

Solution :-

1)

$\sf\dashrightarrow x^3+3x^2+3x+28$

$\sf\dashrightarrow x^3-x^2+4x^2+7x-4x+28$

$\sf\dashrightarrow x^3-x^2+7x+4x^2-4x+28$

$\sf\dashrightarrow Taking\;x\;as\;common$

$\sf\dashrightarrow x(x^2-x+7)+4(x^2-x+7)\bigg(x^2=x\times x\bigg)$

$\sf\dashrightarrow (x^2-x+7)(x+4)=0$

2)

$\sf\dashrightarrow x^2+14x+48$

$\sf\dashrightarrow x^2+(8x+6x)+48$

$\sf\dashrightarrow x^2+8x+6x+48$

$\sf\dashrightarrow x(x+8)+6(x+8)$

$\sf\dashrightarrow (x+8)(x+6)$

3)  

$\sf\dashrightarrow x^4+16$

$\sf\dashrightarrow (x^2)^2+(2^2)^2$

$\sf\dashrightarrow (x^2)^2+(4)^2+2x^2\times 4+(4)^2-8x^2$

$\sf\dashrightarrow (x^2)^2+16+8x^2+16-8x^2$

$\sf\dashrightarrow (x^2+4)^2-(2\sqrt{2}x)^2$

$\sf\dashrightarrow (x^2 + 2\sqrt{2x} + 4)(x^2 - 2\sqrt{2x }+ 4)$