Question 4: Factorise:

(i) `12x^2 - 7x + 1`

**Answer:** By Splitting the middle term method:

Step – 1 – Multiply coefficient of x^{2} i.e. 12 and constant term 1

By multiplication of 12 and 1 we get 12

Step – 2 – Find out the probable prime factors of 12

Prime factors of 12 are (2 and 6) and (3 and 4)

Stepe 3 – Now take that prime factor, sum of which becomes equal to middle term of the given polynomial, i.e. coefficient of x which is equal to – 7

Here, 2 + 6 ≠ 7

3 + 4 = 7

So, we will consider the factor (3 and 4) of 12

Now, arrange the given polynomial as given below:

Given, `12x^2 - 7x + 1`

`= 12x^2 - (3x + 4x) + 1`

(Because `3x + 4x = 7x`)

`= 12x^2 - 3x – 4x + 1`

By taking 3x and -1 as common, we get:

`= 3x(4x – 1) – 1(4x – 1)`

By taking (4x – 1) as common, we get:

`= (3x – 1)(4x – 1)`

Factors of given polynomial `= (3x – 1)` and `(4x – 1)`

(ii) `2x^2 + 7x + 3`

**Answer:** Given, `2x^2 + 7x + 3`

Coefficient of `x^2 = 2`

Constant term = 2

Thus, ab = 6

Prime factors of 6 = 1 and 6

Now, given polynomial can be written as follows:

`2x^2 + x + 6x + 3`

`= x(2x + 1) + 2(2x + 1)`

`= (2x + 1)(x + 3)`

Thus, factor of given polynomial `= (2x + 1)(x + 3)`

(iii) `6x^2 + 5x – 6`

**Answer:** Given, `6x^2 + 5x – 6`

Here, a = 6 and b = 6

Therefore, ab = 36

Factor of 36 = 9 and 4

Thus, given polynomial can be arranged as follows:

`6x^2 + 9x – 4x – 6`

`= 3x(2x + 3) – 2(2x + 3)`

`= (2x + 3)(3x – 2)`

(iv) `3x^2 - x – 4`

**Answer:** Given, `3x^2 - x – 4`

Here, ab = 12

Factor of 12 = 3 and 4

Thus, given polynomial can be arranged as follows:

`3x^2 + 3x – 4x – 4`

`= 3x(x + 1) – 4(x + 1)`

`= (x + 1)(3x – 4)`

Thus, factor of given polynomial `= (x + 1)(3x - 4)`

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