Question

factorise:

x3+13x2+32x+20 (by remainder theorem)

Answer 1

As no specific factor is given here, so we have to use trial method.

So, let ( x + 1) be factor of f(x)
:. x + 1 = 0
or, x = -1
We know, f(x) = x3 + 13x2 + 32x + 20
f(-1) = (-1)3 + 13 × (-1)2 + 32 × (-1) + 20
= -1 + 13 -32 + 20
= 33 - 33
= 0 [The remainder is 0]
Therefore, ( x + 1) is a factor of f(x).

Note: In this expression, ( x + 1) become the factor, but this does not happen in every case. So make sure to repeat the same by taking the successive variable i.e. 2 or any other . Make sure that you keep on trying until the calculation gives the remainder 0.

Hope this helps you.

Answer 2

x^3+13x^2+32x+20= x^3 + x^2+ 12x^2 + 12x + 20x + 20

=x^2(x+1) + 12x(x+1) +20(x+1)

=(x^2 + 12x + 20)(x+1)

=(x^2 + 10x + 2x + 20)(x + 1)

={x(x + 10) + 2(x + 10)}(x + 1)

=(x+1)(x+2)(x+10)