Question

Factorise; a) 12x²−x−6

(b) x⁴−9y⁴

Answer 1

Step-by-step explanation:

(a)

12x2−x−6

=12x2−9x+8x−6

Now we can factor with grouping.

=(12x2−9x)+(8x−6)

=3x(4x−3)+2(4x−3)

=(3x+2)(4x−3)

(b)

Given, x⁴ - 9y⁴

= ( x² )² - ( 3y² )²

Using identity

a² - b² = ( a + b ) ( a - b ) }

= ( x² + 3y² ) ( x² - 3y² )

Again using the identity

= ( x² + 3y² ) [ ( x )² - ( 3y )² ]

= ( x² + 3y² ) ( x + 3y ) ( x - 3y )

Answer 2

My favorite method is the ac method. When you have a trinomial of the form ax2+bx+c, you can start by looking for 2 numbers that multiply to the same value as ac, but add to the same value as b.

For this problem, we have 12x2−x−6, so we need 2 numbers that multiply to the same value as 12(−6)=−72, but the 2 numbers also need to add to the same value as −1.

Since the product of the 2 numbers is negative, that means one of the 2 numbers will have a negative sign, and one will not. Since the negative number plus the positive number equals −1, the number with the negative sign must be very close to the positive number. Therefore, I will seek out 2 numbers that are very close to each other that can multiply to 72.

With a bit of trial and error, I determined that 8 and 9 are 2 numbers that multiply to 72, but are also very close to each other. Therefore, the 2 numbers that multiply to −72 and add to −1 are −9 and 8.

After determining the 2 numbers, you split

up the middle term of the trinomial using those numbers.

12x2−x−6  

12x2−9x+8x−6  

You see this is the same thing, but we have split the middle term into 2 separate terms. Now we can factor with grouping.

(12x2−9x)+(8x−6)  

3x(4x−3)+2(4x−3)  

(3x+2)(4x−3)  

If you multiply that out, you will see that equals the original expression.