Factorisation

Exercise 14.3

(e) `5pq(p^2 – q^2) ÷ 2p (p + q)`

Answer: `5pq(p^2 – q^2) ÷ 2p (p + q)`

Using `(a +b) (a – b) = a^2 – b^2`; the equation can be written as follows:

`= 5pq (p + q) (p – q) ÷ 2p (p + q)`

`= 5q(p – q) ÷ 2`

(f) `12xy (9x^2 – 16y^2) ÷ 4xy (3x + 4y)`

Answer: `12xy (9x^2 – 16y^2) ÷ 4xy (3x + 4y)`
This can be solved as previous question;
`= 3 (3x + 4y) (3x – 4y) ÷ (3x + 4y)`
`= 3 (3x – 4y)`

(g) `39y^3(50y^2 – 98) ÷ 26y^2 (5y + 7)`

Answer: `39y^3(50y^2 – 98) ÷ 26y^2 (5y + 7)`
`= 2 xx 39y^3(25y^2 – 49) ÷ 26y^2 (5y + 7)`
`= 3y (25y^2 – 49) ÷ (5y + 7)`
`= 3y (5y + 7) (5y – 7) ÷ (5y + 7)`
`= 3y (5y – 7)`

Exercise 14.4

Find and correct the errors in the following mathematical statements:

Question 1: `4(x – 5) = 4x – 5`

Answer: `4(x – 5) = 4x – 20`

Question 2: `x(3x + 2) = 3x^2 + 2`

Answer: `= 3x^2 + 2x`

Question 3: `2x + 3y = 5xy`

Answer: 2x + 3y
Here; x and y are as different as chalk and cheese and hence cannot be added together.

Question 4: `x + 2x + 3x = 5x`

Answer: = 6x
It is like adding one apple, two apples and three apples.

Question 5: `5y + 2y + y – 7y = 0`

Answer: `= y`

Question 6: `3x + 2x = 5x^2`

Answer: `= 5x`

Question 7: `(2x)^2 + 4(2x) + 7 = 2x^2 + 8x + 7`

Answer: `= 4x^2 + 8x + 7`

Question 8: `(2x)^2 + 5x = 4x + 5x = 9x`

Answer: `4x^2 + 5x`

Question 9: `(3x + 2)^2 = 3x^2 + 6x + 4`

Answer: Using `(a + b)^2 = a^2 + 2ab + b^2`;
`= 9x^2 + 12x + 4`

Question 10: Substituting `x = - 3` in

(a) `x^2 + 5x + 4` gives `(-3)^2 + 5 ( - 3) + 4 = 9 + 2 + 4 = 15`

Answer: `= (-3)^2 + 5( - 3) + 4`
`= 9 – 15 + 4 = - 2`

(b) `x^2 – 5x + 4` gives `(-3)^2 – 5 (- 3) + 4 = 9 – 15 + 4 = - 2`

Answer: `(- 3)^2 – 5 ( - 3) + 4`
`= 9 + 15 + 4 = 28`

Answer: `( - 3)^2 + 5 ( - 3)`
`= 9 – 15 = - 6`

Question 11: `(y – 3)^2 = y^2 – 9`

Answer: `= y^2 – 6y + 9`

Question 12: `(z + 5)^2 = z^2 + 25`

Answer: `= z^2 + 10z + 25`

Question 13: `(2a + 3b) ( a – b) = 2a^2 – 3b^2`

Answer: `= 2a^2 – 2ab + 3ab – 3b^2`
`= 2a^2 + ab – 3b^2`

Question 14: `(a + 4)(a + 2) = a^2 + 8`

Answer: `= a^2 + 2a + 4a + 8 = a^2 + 6a + 8`

Question 15: `(a – 4)(a – 2) = a^2 – 8`

Answer: `a^2 – 2a – 4a + 8`
`= a^2 – 6a + 8`

Question 16: `3x^2 ÷ 3x^2 = 0`

Answer: = 1

Question 17: `(3x^2 + 1) ÷ 3x^2 = 1 + 1 = 2`

Answer: `= 1 + (1)/(3x^2)`

Question 18: `3x ÷ (3x + 2) = ½`

Answer: `= 3x ÷ (3x + 2)`

Question 19: `3 ÷ (4x + 3) = 1/4x`

Answer: `= 3 ÷ (4x + 3)`

Question 20: `(4x + 5) ÷4x = 5`

Answer: `= 1 + (5)/(4x)`

Question 21: `(7x + 5) ÷ 5 = 7x`

Answer: `= (7x)/(5) + 1`