Factorisation

Exercise 14.3

Question 1: Carry out the following divisions.

(i) `28x^4 ÷ 56x^4`

Answer: `28x^4 ÷ 56x^4`
`= 1/2 x`

(ii) `–36y^2 ÷ 9y^2`

Answer: `-4y`

(iii) `66pq^2r^3 ÷ 11qr^2`

Answer: `6pqr`

(iv) `34x^3y^3z^3 ÷ 51xy^2z^3`

Answer: `2/3x^2y`

(v) `12a^8b^8 ÷ (– 6a^6b^4`)

Answer: `-2a^2b^4`


Question 2: Divide the given polynomial by the given monomial.

(i) `(5x^2 – 6x) ÷ 3x`

Answer: `(5x)/(3) - 2`

(ii) `(3y^8 – 4y^6 + 5y^4) ÷ y^4`

Answer: `3y^4-4y^2+5`

(iii) `8(x^3y^2z^2 + x^2y^3z^2 + x^2y^2z^3) ÷ 4x^2y^2z^2`

Answer: `2(x+y+z)`

(iv) `(x^3 + 2x^2 + 3x) ÷ 2x`

Answer: `1/2x^2+2x+3/2`

(v) `(p^3q^6 – p^6q^3) ÷ p^3q^3`

Answer: `q^3 - p^3`

Question 3: Work out the following divisions.

(i) `(10x – 25) ÷ 5`

Answer: `2x-5`

(ii) `(10x – 25) ÷ (2x – 5)`

Answer: 5

(iii) `10y(6y + 21) ÷ 5(2y + 7)`

Answer: `2y \xx 3 = 6y`

(iv) `9x^2y^2 (3z – 24) ÷ 27xy(z – 8)`

Answer: `1/3xy\xx\3=xy`

(v) `96abc(3a – 12) (5b – 30) ÷ 144(a – 4) (b – 6)`

Answer: `2/3abc\xx3xx5=10abc`


Question 4: Divide as directed:

(a) `5(2x + 1)(3x + 5) ÷ (2x + 1)`

Answer: `5(2x + 1)(3x + 5) ÷ (2x + 1)`
`= 5(3x + 5) = 15x + 25`

(b) `26xy(x + 5) (y – 4) ÷ 13x (y -4)`

Answer: `26xy(x + 5) (y – 4) ÷ 13x (y -4)`
`= 2y(x + 5)(y – 4) ÷(y – 4)`
`= 2Y (x + 5) = 2xy + 10y`

(c) `52pqr(p + q) (q + r)( r + p) ÷ 104 pq(q + r)( r + p)`

Answer: `52pqr(p + q) (q + r)( r + p) ÷ 104 pq(q + r)( r + p)`
`= r (p + q) (q + r) ( r + p) ÷ 2 (q + r) (r + p)`
`= r (p + q) ÷ 2`

(d) `20 ( y + 4) (y^2 + 5y + 3) ÷ 5 (y + 4)`

Answer: `20 ( y + 4) (y^2 + 5y + 3) ÷ 5 (y + 4)`
`= 4 (y + 4) (y^2 + 5y + 3) ÷ (y + 4)`
`= 4(y^2 + 5y + 3)`

(e) `x ( x + 1) ( x + 2) (x + 3) ÷ x( x + 1)`

Answer: `x ( x + 1) ( x + 2) (x + 3) ÷ x( x + 1)`
`= (x + 2)(x + 3)`

Question 5: Factorise the expressions and divide them as directed

(a) `(y^2 + 7y + 10) ÷ (y + 5)`

Answer: `(y^2 + 7y + 10) ÷ (y + 5)`
Here, dividend can be factorised by splitting the middle term as follows:
`= (y^2 + 5y + 2y + 10) ÷ (y + 5)`
`= [y (y + 5) + 2 (y + 5) ] ÷ (y + 5)`
`= (y + 2) (y + 5) ÷ (y + 5)`
`= y + 2`

(b) `(m^2 – 14m – 32) ÷ (m + 2)`

Answer: `(m^2 – 14m – 32) ÷ (m + 2)`
Here, dividend can be factorised by splitting the middle term as follows:
`= (m^2 – 16m + 2m – 32) ÷ (m + 2)`
`= [ m(m – 16) + 2(m – 16) ] ÷ (m + 2)`
`= (m + 2) (m – 16) ÷ (m +2)`
`= m – 16`

(c) `5p^2 – 25p + 20) ÷ (p – 1)`

Answer: `5p^2 – 25p + 20) ÷ (p – 1)`
Here, dividend can be factorised by splitting the middle term as follows:
`= (5p^2 – 5p – 20p + 20) ÷ (p - 1)`
`= [5p(p – 1) – 20(p – 1) ] ÷ (p – 1 )`
`= (5p – 20) (p – 1) ÷ (p – 1)`
`= 5p – 20`

(d) `4yz(z^2 + 6z – 16) ÷ 2y (z + 8)`

Answer: `4yz(z^2 + 6z – 16) ÷ 2y (z + 8)`
`= 2z(z^2 + 8z – 2z – 16) ÷ (z + 8)^2`
`= 2z [ z(z + 8) – 2(z + 8) ] ÷ (z + 8)`
`= 2z (z – 2) (z + 8) ÷(z + 8)`
`= 2z (z – 2)`



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