Factorisation
Exercise 14.1
Question 1: Find the common factors of the given terms.
(i) 12x, 36
Answer: `12x=2xx2xx3xx\x`
`36=2xx2xx3xx3`
Here, common factor `= 2xx3=6`
(ii) 2y, 22xy
Answer: `2y=2xx\y`
`22y=2xx11xx\y`
Common factor = 2
(iii) 14 pq, 28 p2q2
Answer: `14pq=2xx7xx\p\xx\q`
`28p^2q^2=2xx2xx7xx\p\xx\p\xx\q\xx\q`
Common factor `=2xx7xx\p\xx\q`
(iv) 2x, 3x², 4
Answer: `2x=2xx\x`
`3x^2=3xx\x\xx\x`
`4=2xx2`
Common factor = 1
There is no common factor other than unity, so common factor= 1
(v) 6 abc, 24ab², 12 a²b
Answer: `6abc=2xx3xx\a\xx\b\xx\c`
`24ab^2=2xx2xx2xx3xx\a\xx\b\xx\b`
`12a^2b=2xx2xx3xx\a\xx\a\xx\b`
Common Factor `=2xx3xx\a\xx\b`
(vi) 16 x³, – 4x², 32x
Answer: `16x^3=2xx2xx2xx2xx\a\xx\a\xx\a`
`-4x^2=-2xx2xx\x\xx\x`
`32x=2xx2xx2xx2xx2xx\x`
Common factor `=2x`
(vii) 10 pq, 20qr, 30rp
Answer: `10pq=2xx5xx\p\xx\q`
`20qr=2xx2xx4xx\q\xx\r`
`30rp=2xx3xx5xx\r\xx\p`
Common factor `=2xx5=10`
(viii) 3x² y³, 10x³ y²,6 x² y²z
Answer: `3x^2y^3=3xx\x\xx\x\xx\y\xx\y\xx\y`
`10x^3y^2=2xx5xx\x\xx\x\xx\x\xx\y\xx\y`
`x^2y^2z=x\xx\x\xx\y\xxy\xx\z`
Common factor`=x^2y^2`
Question 2: Factorise the following expressions.
(i) 7x – 42
Answer: `7x-42`
`=7xx\x-7xx6=7(x-6)`
(ii) 6p – 12q
Answer: `6p-12q`
`=6xx\p-6xx2q=6(p-2q)`
(iii) 7a² + 14a
Answer: `7a^2+14a`
`=7xx\a\xx\a+7xx2xx\a=7a(a+2)`
(iv) – 16 z + 20 z³
Answer: `-16z+20z^3`
`=4xx4xx\z+4xx5xx\z\xx\z\xx\z`
`=4z(-4+5z\xx\z)=4z(5z^2-4)`
(v) 20 l² m + 30 a l m
Answer: `20l^2m+30alm`
`=2xx2xx5xx\l\xx\l\xx\m+2xx3xx5xx\a\xx\l\xx\m`
`=2lm(10l+15a)`
(vi) 5 x² y – 15 xy²
Answer: `5x^2y-15xy^2`
`=5xx\x\xx\x\xx\y-3xx5xx\x\xx\y\xx\y`
`=5x(xy-3y^2)`
(vii) 10 a² – 15 b² + 20 c²
Answer: `10a^2-15b^2+20c^2`
`=2xx5xx\a\xx\a-3xx5xx\b\xx\b+4xx5xx\c\xx\c`
`=5(2a^2-3b^2+4c^2)`
(viii) – 4 a² + 4 ab – 4 ca
Answer: `-4a^2+4ab-4ca`
`=-4xx\a\xx\a+4xx\a\xx\b-4xx\c\xx\a`
`=-4a(a-b+c)`
(ix) x² y z + x y²z + x y z²
Answer: `x^2yz+xy^2z+xyz^2`
`=x\xx\x\xx\y\xx\z+x\xx\y\xx\y\xx\z+x\xx\y\xx\z\xx\z`
`=xyz(x+y+z)`
(x) a x² y + b x y² + c x y z²
Answer: `ax^yz+bxy^2z+cxyz^2`
`=a\xx\x\xx\x\xx\y\xx\z+b\xx\x\xx\y\xx\y\xx\z+c\xx\x\xx\y\xx\z\xx\z`
`=xyz(ax+by+cz)`
Question 3: Factorise.
(i) x² + x y + 8x + 8y
Answer: `x^2+xy+8x+8y`
`=x\xx\x+x\xx\y+8xx\x+8xx\y`
`=x(x+y)+8(x+y)`
`=(x+8)(x+y)`
(ii) 15 xy – 6x + 5y – 2
Answer: `15xy-6x+5y-2`
`=3xx5xx\x\xx\y-2xx3xx\x+5xx\y-2`
`=3x(5y-2)+1(5y-2)`
`=(3x+1)(5y-2)`
(iii) ax + bx – ay – by
Answer: `ax+bx-ay-by`
`=x(a+b)-y(a+b)`
`=(x-y)(a+b)`
(iv) 15 pq + 15 + 9q + 25p
Answer: `15pq+15+9q+25p`
`=5xx3xx\pq+5xx3+3xx3q+5xx5p`
`=5xx3pq+3xx3q+5xx5p+5xx3`
`=3q(5p+3)+5(5p+3)`
`=(3q+5)(5p+3)`
(v) z – 7 + 7 x y – x y z
Answer: `z-7+7xy-xyz`
`=z-xyz-7+7xy`
`=z(1-xy)-7(1-xy)`
`=(z-7)(1-xy)`