Question

Factorise the following expressions

(i) 7x − 42

(ii) 6p − 12q

(iii) 7a² + 14a

(iv) −16z + 20z³

(v) 20l²m + 30 alm

(vi) 5x²y − 15xy²

(vii) 10a² − 15b² + 20c²

(viii) −4a² + 4ab − 4 ca

(ix) x²yz + xy²z + xyz²

(x) ax²y + bxy² + cxyz

Answer 1

(i) 7x = 7 × x

42 = 2 × 3 × 7

The common factor is 7.

∴ 7x − 42 = (7 × x) − (2 × 3 × 7) = 7 (x − 6)

(ii) 6p = 2 × 3 × p

12q = 2 × 2 × 3 × q

The common factors are 2 and 3.

∴ 6p − 12q = (2 × 3 × p) − (2 × 2 × 3 × q)

= 2 × 3 [p − (2 × q)]

= 6 (p − 2q)

(iii) 7a² = 7 × a × a

14a = 2 × 7 × a

The common factors are 7 and a.

∴ 7a² + 14a = (7 × a × a) + (2 × 7 × a)

= 7 × a [a + 2] = 7a (a + 2)

(iv) 16z = 2 × 2 × 2 × 2 × z

20z³ = 2 × 2 × 5 × z × z × z

The common factors are 2, 2, and z.

∴ −16z + 20z³ = − (2 × 2 × 2 × 2 × z) + (2 × 2 × 5 × z × z × z)

= (2 × 2 × z) [− (2 × 2) + (5 × z × z)]

= 4z (− 4 + 5z²)

(v) 20l²m = 2 × 2 × 5 × l × l × m

30alm = 2 × 3 × 5 × a × l × m

The common factors are 2, 5, l, and m.

∴ 20l²m + 30alm = (2 × 2 × 5 × l × l × m) + (2 × 3 × 5 × a × l × m)

= (2 × 5 × l × m) [(2 × l) + (3 × a)]

= 10lm (2l + 3a)

(vi) 5x²y = 5 × x × x × y

15xy² = 3 × 5 × x × y × y

The common factors are 5, x, and y.

∴ 5x²y − 15xy² = (5 × x × x × y) − (3 × 5 × x × y × y)

= 5 × x × y [x − (3 × y)]

= 5xy (x − 3y)

(vii) 10a² = 2 × 5 × a × a

15b² = 3 × 5 × b × b

20c² = 2 × 2 × 5 × c × c

The common factor is 5.

10a² − 15b² + 20c² = (2 × 5 × a × a) − (3 × 5 × b × b) + (2 × 2 × 5 × c × c)

= 5 [(2 × a × a) − (3 × b × b) + (2 × 2 × c × c)]

= 5 (2a² − 3b² + 4c²)

(viii) 4a² = 2 × 2 × a × a

4ab = 2 × 2 × a × b

4ca = 2 × 2 × c × a

The common factors are 2, 2, and a.

∴ −4a² + 4ab − 4ca = − (2 × 2 × a × a) + (2 × 2 × a × b) − (2 × 2 × c × a)

= 2 × 2 × a [− (a) + b − c]

= 4a (−a + b − c)

(ix) x²yz = x × x × y × z

xy²z = x × y × y × z

xyz² = x × y × z × z

The common factors are x, y, and z.

∴ x²yz + xy²z + xyz² = (x × x × y × z) + (x × y × y × z) + (x × y × z × z)

= x × y × z [x + y + z]

= xyz (x + y + z)

(x) ax²y = a × x × x × y

bxy² = b × x × y × y

cxyz = c × x × y × z

The common factors are x and y.

ax²y + bxy² + cxyz = (a × x × x × y) + (b × x × y × y) + (c × x × y × z)

= (x × y) [(a × x) + (b × y) + (c × z)]

= xy (ax + by + cz)