Question

use synthetic division to determine which of the following are the real zeroes of the function f(x) = 5x^3 + 8x^2 + x - 2

Answer 1

Always try putting x=1 and x=-1 to see if we can find a zero.
If x=1 we get 5+8+1-2 which is not zero.
If we change the signs of the odd powers of x we get -5+8-1-2 which is zero so x=-1 is a zero and x+1 must therefore be a factor. Now we can use synthetic division to find the quadratic:

-1 | 5 8 1 -2
5 -5 -3 2
————
5 3 -2 | 0 = 5x²+3x-2 = (5x-2)(x+1).

Therefore the real zeroes are -1, 2/5, since -1 is a duplicate zero (because the cubic factorises into (5x-2)(x+1)²).