Question

Factorise : (a² - 3a) (a² - 3a + 7) + 10

Answer 1

(a² - 3a) (a² - 3a + 7) + 10

Let us assume , a² - 3a = x
Then, our polynomial becomes,
( a² - 3a )( a² - 3a + 7 ) + 10
= x( x + 7 ) + 10
= x² + 7x + 10
= x² + 5x + 2x + 10
= x( x + 5 ) + 2 ( x + 5 )
= ( x + 5 )( x + 2 )

By resubstituting the value of x,
= (a² - 3a + 5)( a² - 3a + 2 )

Now, a² - 3a + 5 will have no factor as discriminant is -11 that is less than 0.

And,

∴ a² - 3a + 2 = a² - 2a - a + 2 = a( a - 2) - 1(a - 2) = (a - 1)(a - 2)

So, factor of given polynomial are,

a² - 3a + 2 = a² - 2a - a + 2

= (a² - 3a + 5)(a - 1)(a - 2)