Factorisation — Exercise 4.1

Factorise the following:

8xy³ + 12x²y²

Answer

H.C.F. of 8xy³ and 12x²y² is 4xy².

∴ 8xy³ + 12x²y² = 4xy²(2y + 3x).

Factorise the following:

15ax³ - 9ax².

Answer

H.C.F. of 15ax³ and 9ax² is 3ax².

∴ 15ax³ - 9ax² = 3ax²(5x - 3).

Factorise the following:

21py² - 56py

Answer

H.C.F. of 21py² and 56py is 7py.

∴ 21py² - 56py = 7py(3y - 8).

Factorise the following:

4x³ - 6x².

Answer

H.C.F. of 4x³ and 6x² is 2x².

∴ 4x³ - 6x² = 2x²(2x - 3).

Factorise the following:

2πr² - 4πr

Answer

H.C.F. of 2πr² and 4πr is 2πr.

∴ 2πr² - 4πr = 2πr(r - 2).

Factorise the following:

18m + 16n

Answer

H.C.F. of 18m and 16n is 2.

∴ 18m + 16n = 2(9m + 8n).

Factorise the following:

25abc² - 15a²b²c

Answer

H.C.F. of 25abc² and 15a²b²c is 5abc.

∴ 25abc² - 15a²b²c = 5abc(5c - 3ab).

Factorise the following:

28p²q²r - 42pq²r²

Answer

H.C.F. of 28p²q²r and 42pq²r² is 14pq²r.

∴ 28p²q²r - 42pq²r² = 14pq²r(2p - 3r).

Factorise the following:

8x³ - 6x² + 10x

Answer

H.C.F. of 8x³, 6x² and 10x is 2x.

∴ 8x³ - 6x² + 10x = 2x(4x² - 3x + 5).

Factorise the following:

14mn + 22m - 62p

Answer

H.C.F. of 14mn, 22m and 62p is 2.

∴ 14mn + 22m - 62p = 2(7mn + 11m - 31p).

Factorise the following:

18p²q² - 24pq² + 30p²q

Answer

H.C.F. of 18p²q², 24pq² and 30p²q is 6pq.

∴ 18p²q² - 24pq² + 30p²q = 6pq(3pq - 4q + 5p).

Factorise the following:

27a³b³ - 18a²b³ + 75a³b²

Answer

H.C.F. of 27a³b³, 18a²b³ and 75a³b² is 3a²b².

∴ 27a³b³ - 18a²b³ + 75a³b² = 3a²b²(9ab - 6b + 25a).

Factorise the following:

15a(2p - 3q) - 10b(2p - 3q)

Answer

H.C.F. of 15a(2p - 3q) and 10b(2p - 3q) is 5(2p - 3q).

∴ 15a(2p - 3q) - 10b(2p - 3q) = 5(2p - 3q)(3a - 2b).

Factorise the following:

3a(x² + y²) + 6b(x² + y²).

Answer

H.C.F. of 3a(x² + y²) and 6b(x² + y²) is 3(x² + y²).

∴ 3a(x² + y²) + 6b(x² + y²) = 3(x² + y²)(a + 2b).

Factorise the following:

6(x + 2y)³ + 8(x + 2y)²

Answer

H.C.F. of 6(x + 2y)³ and 8(x + 2y)² is 2(x + 2y)².

∴ 6(x + 2y)³ + 8(x + 2y)² = 2(x + 2y)²(3(x + 2y) + 4) = 2(x + 2y)²(3x + 6y + 4).

Factorise the following:

14(a - 3b)³ - 21p(a - 3b)

Answer

H.C.F. of 14(a - 3b)³ and 21p(a - 3b) is 7(a - 3b).

∴ 14(a - 3b)³ - 21p(a - 3b) = 7(a - 3b)[2(a - 3b)² - 3p].

Factorise the following:

10a(2p + q)³ - 15b(2p + q)² + 35(2p + q)

Answer

H.C.F. of 10a(2p + q)³, 15b(2p + q)² and 35(2p + q) is 5(2p + q).

∴ 10a(2p + q)³ - 15b(2p + q)² + 35(2p + q) = 5(2p + q)[2a(2p + q)² - 3b(2p + q) + 7].

Factorise the following:

x(x² + y² - z²) + y(-x² - y² + z²) - z(x² + y² - z²).

Answer

The above expression:

x(x² + y² - z²) + y(-x² - y² + z²) - z(x² + y² - z²)

can be written as

x(x² + y² - z²) + y(-1)(x² + y² - z²) - z(x² + y² - z²)
= x(x² + y² - z²) - y(x² + y² - z²) - z(x² + y² - z²).

H.C.F. of x(x² + y² - z²), y(x² + y² - z²) and z(x² + y² - z²) is (x² + y² - z²).

∴ x(x² + y² - z²) + y(-x² - y² + z²) - z(x² + y² - z²) = (x² + y² - z²)(x - y - z).