Synthetic Division Verification--ASAP!!!?

Question

Use synthetic Division to decide whether the given number is a zero of the given polynomial>

3:f(x)=9x^3+x^2+4x-5

Answer 1

Using synthetic division we have:

x - 3)9x^3 + x^2 + 4x - 5(9x^2 + 28x + 88
-------9x^3 - 27x^2....... subtracting
----------------28x^2 + 4x......bringing down +4x from above
----------------28x^2 - 84x.........subtracting
---------------------------88x - 5.....bringing down -5 from above
---------------------------88x - 264...subtracting
----------------------------------259
In this synthetic division, if 3 is a zero of the given polynomial, then (x - 3) would be a factor of the given polynomial. So we divide the given polynomial by (x - 3).
step 1: we multiply (x - 3)by 9x^2 and get 9x^3 - 27x^2 which we subtract from the given polynomial and get 28x^2 we bring down 4x to join this to get (28x^2 + 4x)

step 2:we multiply (x - 3)by 28x and get 28x^2 - 84x which we subtract from 28x^2 + 4x and get 88x we bring down -5 to join this and get (88x - 5)

step 3: we multiply (x - 3)by 88 and get 88x - 264 which we subtract from 88x - 5 and get 259.

The factors we used in multiplying (x - 3) in steps 1 to 3 is written above to the right of the polynomial and the remainder in the synthetic division is the last term in the expressions above.

Answer 2

OOPS! Missed your 3 earlier! Sorry.
Let's see what happens in synthetic division here:
3 9 1 4 -5
____27 84 264_____
9 28 88// 259
Since the remainder is 259, 3 is not a zero.
In fact, if you run this through PARI you will find
that f(x) has no rational root at all and so is irreducible.