Question

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Answer 1

Question

Factorise it

1. a³ - 27
2. 1 - 125y³
3. x^6 - y^6
4. a¹² - b¹²
5. 7m⁴n + 7mn⁴
6. a³ + b³ +c(a² - ab +b²)
7. x³ + y³ + 2x + 2y

Solution

1) a³ + 27

= (a)³ + (3)³

Applying identity

a³ + b³=(a+b)(a²-ab+b²)

= (a + 3) [(a)² + a×3 + (3)²]

= (a + 3) (a² + 3a + 9)

_____________________

2) 1 - 125y³

= (1)³ - (5y)³

Applying identity

a³ - b³ = (a+b)(a² + ab +b²)

= (1 - 5y) [(1)² + 1×5y + (5y)²]

= (1 - 5y) (1 + 5y + 25y²)

_____________________

3) x^6 - y^6

= (x³)² - (y³)²

Again applying identity

a² - b² = (a+b)(a - b)

= (x³ + y³) (x³ - y³)

Again applying identity

a³ + b³ = (a+b)(a² - ab +b²)

a³ - b³ = (a+b)(a² + ab +b²)

= [(x + y)(x² - xy + y²)] [(x - y)(x² + xy + y²)]

= (x + y)(x - y)(x² - xy + y²)(x² + xy + y²)

_____________________

4) a¹² - b¹²

= (a^6)² - (b^6)²

Applying identity

a² - b² = (a + b)(a - b)

= (a^6 + b^6)(a^6 - b^6)

= [(a²)³ + (b²)³] [(a³)²- (b³)²]

Applying identity

a³ + b³ = (a+b)(a² - ab +b²)

a² - b² = (a + b)(a - b)

= [(a² + b²)(a⁴ - a²b² + b⁴)] [(a³ - b³)(a³ + b³)]

Applying identity

a³ - b³ = (a+b)(a² + ab +b²)

a³ + b³ (a+b)(a² - ab +b²)

= [(a² + b²)(a⁴ - a²b² + b⁴)] [{(a-b)³(a² + ab + b²)}{(a+b)(a² - ab + b²)}]

= (a² + b²)(a-b)(a+b)(a⁴ - a²b² + b⁴)(a² + ab + b²)(a² - ab + b²)

_____________________

5) 7m⁴n - 7mn⁴

Take 7mn as common

= 7mn(m³ - n³)

Again applying identity

a³- b³ = (a+b)(a² +ab +b²)

= 7mn[(m-n)(m² +mn+ n²)]

_____________________

6) a³ + b³ + c(a² - ab + b²)

Applying identity

a³ + b³ = (a+b)(a² - ab +b²)

= (a+b)(a² - ab+b²)+c(a² - ab + b²)

Take (a² - ab + b²) as a common

= (a² - ab + b²)(a + b + c)

_____________________

7) x³ + y³ + 2x + 2y

Again applying identity

a³ + b³ = (a+b)(a² - ab +b²)

= (x+y)(x² - xy + y²)+2(x+y)

Take (x+y) as a common

= (x + y)(x²- xy + y² + 2)

Answer 2

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$\huge\tt{GIVEN:}$

1. a³+27
2. 1- 125y³
3. x^6-y^6
4. a¹² - b¹²
5. 7m⁴n-7mn⁴
6. a³+b³+c(a²-ab+b²)
7. x³+y³+2x+2y

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$\huge\tt{TO~FACTORISE~THEM:}$

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$\huge\tt{SOLUTIONS:}$

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QUESTION.1

↪a³+27

↪a³+3³

↪(a+3)[(a²)+a×3+(3²)]

↪(a+3)(a²+3a+9)

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QUESTION.2

↪1- 125y³

↪(1³)-(5y³)

↪(1-5y)[1²+1×5y+5y²]

↪(1-5y)(1+5y+25y²)

_______________________________

QUESTION.3

↪x^6-y^6

↪(x³)²-(y³)²

↪(x³+y³)(x³-y³)

↪[(x+y)(x²-xy+y²)][(x-y)(x²+xy+y²)]

↪(x+y)(x-y)(x²-xy+y²)(x²+xy+y²)

_______________________________

QUESTION.4

↪a¹²-b¹²

↪(a^6)²-(b^6)²

↪(a^6+b^6)(a^6-b^6)

↪[(a²)³+(b²)³][(a³)²-(b³)²]

↪[(a²+b²)(a⁴-a²b²+b⁴)][(a³-b³)(a³+b³)]

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QUESTION.5

↪7m⁴n-7mn⁴

↪7mn(m³-n³)

↪7mn[(m-n)(m²+mn+n²)]

_______________________________

QUESTION.6

↪a³+b³+c(a²-ab+b²)

↪(a+b)(a²-ab+b²)+c(a²-ab+b²)

↪(a²-ab+b²)(a+b+c)

_______________________________

QUESTION.7

↪x³+y³+2x+2y

↪(x+y)(x²-xy+y²)+2(x+y)

↪(x+y)(x²-xy+y²+2)

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