Question

Factorise the following by regrouping the terms.
a) 6xy-y^2+12xz-2yz
b) b^2-ab(1-a)-a^3
c) a^2-a(x+2y)+2xy
d) x^3-2x^2y+3xy^2-6y^3
e) xy(a^2+1)-a(x^2+y^2)​

Answer 1

Question :-

Factorise the following by regrouping the terms.

a) 6xy  - y² + 12xz - 2yz

b) b² - ab(1 - a) - a³

c) a² - a(x + 2y) + 2xy

d) x³ - 2x²y + 3xy² - 6y³

e) xy(a² + 1)-a(x² + y²)

Solution :-

a) $\sf{6xy-y^2+12xz-2yz}$

$\sf{= y(6x-y)+2z(6x-y) }$

$\sf{= (6x-y)(y+2z)}$

◘ The factorised form is (6x - y)(y + 2z).

_________________

b) $\sf{ b^2-ab(1-a)-a^3}$

$\sf{=b^2-ab+a^2b-a^3 }$

$\sf{=b(b-a)+a^2(b-a)}$

$\sf{=(b-a)(b+a^2)}$

◘ The factorised form is (b - a)(b + a²).

_________________

c) $\sf{a^2-a(x+2y)+2xy}$

$\sf{=a^2-ax-2ay+2xy }$

$\sf{= a(a-x)-2y(a-x)}$

$\sf{= (a-x)(a-2y)}$

◘ The factorised form is (a - x)(a - 2y).

_________________

d) $\sf{x^3-2x^2y+3xy^2-6y^3}$

$\sf{=x^2(x-2y)+3y^2(x-2y) }$

$\sf{= (x-2y)(x^2+3y^2)}$

◘ The factorised form is (x - 2y)(x² + 3y²).

_________________

e) $\sf{xy(a^2+1)-a(x^2+y^2)}$

$\sf{= a^2xy+xy-ax^2-ay^2}$

$\sf{=a^2xy - ax^2 + xy - ay^2}$

$\sf{=ax(ay-x)-y(ay-x)}$

$\sf{=( ax - y )( ay - x )}$

◘ The factorised form is (ax - y)(ay - x).

Answer 2

Step-by-step explanation:

Factorise the following by regrouping the terms.

a) 6xy - y² + 12xz - 2yz

b) b² - ab(1 - a) - a³

c) a² - a(x + 2y) + 2xy

d) x³ - 2x²y + 3xy² - 6y³

e) xy(a² + 1)-a(x² + y²)

\huge \sf {\orange {\underline {\pink{\underline{Solution :-}}}}}

Solution:−

a) \sf{6xy-y^2+12xz-2yz}6xy−y

2

+12xz−2yz

\sf{= y(6x-y)+2z(6x-y) }=y(6x−y)+2z(6x−y)

\sf{= (6x-y)(y+2z)}=(6x−y)(y+2z)

◘ The factorised form is (6x - y)(y + 2z).

_________________

b) \sf{ b^2-ab(1-a)-a^3}b

2

−ab(1−a)−a

3

\sf{=b^2-ab+a^2b-a^3 }=b

2

−ab+a

2

b−a

3

\sf{=b(b-a)+a^2(b-a)}=b(b−a)+a

2

(b−a)

\sf{=(b-a)(b+a^2)}=(b−a)(b+a

2

)

◘ The factorised form is (b - a)(b + a²).

_________________

c) \sf{a^2-a(x+2y)+2xy}a

2

−a(x+2y)+2xy

\sf{=a^2-ax-2ay+2xy }=a

2

−ax−2ay+2xy

\sf{= a(a-x)-2y(a-x)}=a(a−x)−2y(a−x)

\sf{= (a-x)(a-2y)}=(a−x)(a−2y)

◘ The factorised form is (a - x)(a - 2y).

_________________

d) \sf{x^3-2x^2y+3xy^2-6y^3}x

3

−2x

2

y+3xy

2

−6y

3

\sf{=x^2(x-2y)+3y^2(x-2y) }=x

2

(x−2y)+3y

2

(x−2y)

\sf{= (x-2y)(x^2+3y^2)}=(x−2y)(x

2

+3y

2

)

◘ The factorised form is (x - 2y)(x² + 3y²).

_________________

e) \sf{xy(a^2+1)-a(x^2+y^2)}xy(a

2

+1)−a(x

2

+y

2

)

\sf{= a^2xy+xy-ax^2-ay^2}=a

2

xy+xy−ax

2

−ay

2

\sf{=a^2xy - ax^2 + xy - ay^2}=a

2

xy−ax

2

+xy−ay

2

\sf{=ax(ay-x)-y(ay-x)}=ax(ay−x)−y(ay−x)

\sf{=( ax - y )( ay - x )}=(ax−y)(ay−x)

◘ The factorised form is (ax - y)(ay - x).

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