Question

1 Show that (e + 5) is a factor of f (x) = - 22 - 23e + 60 then factor f (x) completely. You will need to use synthetic division for this,

Answer 1

(x+5)(x-4)(x-3)

Explanation:

Divide (e+5) into e³-2e²-23e+60 using polynomial long division (picture attached below). Since it divides in evenly, we can see it is a factor.

So we can say e³-2e²-23e+60 = (e+5)(e²-7e+12)

To complete the factorisation, factorise e²-7e+12.

Note that (x+a)(x+b)=x²+(a+b)x+ab. So find the multiples or 12 which can be added or subtracted to get -7: -4 and -3.

e²-7e+12=(x-4)(x-3)

So e³-2e²-23e+60 = (x+5)(x-4)(x-3)