Factorisation

Exercise 14.2

Question 1: Factorise the following expressions.

(i) `a^2 + 8a + 16`

Answer: This equation can be facorised by using the identity; `(a + b)^2 = a^2 + 2ab + b^2`
Factors `= (a + 4)^2 = (a + 4)(a + 4)`

(ii) `p^2 – 10 p + 25`

Answer: This equation can be factorised by using the identity; `(a – b)^2 = a^2 – 2ab + b^2`
Factors `= (p – 5)^2`

(iii) `25m^2 + 30m + 9`

Answer: `= (5m – 3)^2`

(iv) `49y^2 + 84yz + 36z^2`

Answer: `(7y + 6z)^2`

(v) `4x^2 – 8x + 4`

Answer: `(2x – 2)^2`

(vi) `121b^2 – 88bc + 16c^2`

Answer: `(11b – 4c)^2`

(vii) `(l + m)^2 – 4lm`

Answer: `l^2 + m^2 + 2lm - 4lm`
`l^2+ m^2 - 2lm = (l + m)^2`

(viii) `a^4 + 2a^2b^2 + b^4`

Answer: This can be solved using `(a+b)^2 = a^2+ 2ab + b^2`
Hence, `a^4 + 2a^2b^2 + b^4`
`=(a^2+b^2)^2`


Question 2: Factorise.

(i) `4p^2 – 9q^2`

Answer: This can be factorised by using the equation; `(a + b)(a – b) = a^2 – b^2`
Factors `= (2p + 3q)(2p – 3q)`

(ii) `63a^2 – 112b^2`

Answer: `63a^2 – 112b^2 = 7(9a^2 – 16b^2)`
`= 7(3a + 4b)(3a – 4b)`

(iii) `49x^2 – 36`

Answer: `(7x + 6)(7x – 6)`

(iv) `16x^5 – 144x^3`

Answer: `16x^5-144x^3`
`= x^3(16x^2-144)`
`= x^3(4x+12)(4x-12)`

(v) `(l + m)^2 – (l – m)^2`

Answer: `(l + m + l – m)(l + m – l + m)`
`2l \xx 2m = 4lm`

(vi) `9x^2 y^2 – 16`

Answer: `(3xy + 4)(3xy – 4)`

(vii) `(x^2 – 2xy + y^2) – z^2`

Answer: `(x^2 – 2xy + y^2) – z^2`
`= (x – y)^2 – z^2`
`= (x – y + z)(x – y – z)`

(viii) `25a^2 – 4b^2 + 28bc – 49c^2`

Answer: `25a^2 – 4b^2 + 28bc – 49c^2`
`= (5a)^2 – (2b)^2 + 2 xx 2b \xx 7c – (7c)^2`
`= (5a)^2 – [(2b)^2 – 2 xx 2b\ xx 7c + (7c)^2]`
`= (5a)^2 – (2b – 7c)^2`
This can be further factorised by using `(a + b)(a – b) = a^2 – b^2`
`= (5a + 2b – 7c)(5a – 2b + 7c)`


Question 3: Factorise the expressions.

(i) `ax^2 + bx`

Answer: `x(ax + b)`

(ii) `7p^2 + 21q^2`

Answer: `7(p^2 + 3q^2)`

(iii) `2x^3 + 2xy^2 + 2xz^2`

Answer: `2x^3 + 2xy^2 + 2xz^2`
`= 2x(x^2+y^2+z^2)`

(iv) `am^2 + bm^2 + bn^2 + an^2`

Answer: `a(m^2 + n^2) + b(m^2 + n^2)`
`= (a + b)(m^2 + n^2)`

(v) `(lm + l) + m + 1`

Answer: `l(m + 1) + 1(m + 1)`
`= (l + 1)(m + 1)`

(vi) `y (y + z) + 9 (y + z)`

Answer: `(y + 9)(y + z)`

(vii) `5y^2 – 20y – 8z + 2yz`

Answer: `5y(y + 4) + 2z(y + 4)`
`= (5 + 2z)(y + 4)`

(viii) `10ab + 4a + 5b + 2`

Answer: `5b + 10ab + 2 + 4a`
`= 5b(1 + 2a) + 2(1 + 2a)`
`= (5b + 2)(1 + 2a)`

(ix) `6xy – 4y + 6 – 9x`

Answer: `6xy – 4y + – 9x + 6`
`= 2y (3x – 2) - 3 (3x - 2)`
`= (2y – 3)(3x – 2)`

Question 4: Factorise.

(i) `a^4 – b^4`

Answer: `a^4-b^4 = (a^2+b^2)(a^2-b^2)`

(ii) `p^4 – 81`
`=(p^2+9)(p^2-9)`

(iii) `x^4 – (y + z)^4`

Answer: `x^4 – (y + z)^4`
`= (x^2+(y+z)^2)(x^2-(y+z)^2)`
`= (x^2+(y+z)^2)[(x+y+z)(x-y-z)]`

(iv) `x^4 – (x – z)^4`

Answer: `x^4 – (x – z)^4`
`=(x^2-(x-z)^2)(x^2+(x-z)^2)`
`=[(x+x-z)(x-x+z)][x^2+(x-z)^2]`

(v) `a^4 – 2a^2b^2 + b^4`

Answer: `a^4 – 2a^2b^2 + b^4`
This can be factorised by using the identity; `(a - b)^2 = a^2 – 2ab + b^2`
Factors `= (a^2 – b^2)^2 = (a^2 – b^2)(a^2 – b^2)`

Question 5: Factorise the following expressions.

(i) `p^2 + 6p + 8`

Asnwer: `p^2+6p+8`
`= p(p+6)+8`

(ii) `q^2 – 10q + 21`

Answer: `q^2-10q+21`
`= q(q-10)+21`

(iii) `p^2 + 6p – 16`

Answer: `p^2+6p-16`
`= p(p + 6)- 16`



Copyright © excellup 2014