x^4-6x^3+4x^2+24x-32

Question

Potential rational zeros

Answer 1

Rational zeroes are factors of 32 and can be either positive or negative.

Factors of 32 are 1, 2, 4, 8, 16, 32.

To test if x-1 or x+1 is a factor of the polynomial write the coefficients and place ± in front of the coefficients which have odd powers of x:

x∓6+4±24-32 so we have either 1-6+4+24-32=-9 or 1+6+4-24-32=-45, so neither 1 nor -1 are zeroes of the polynomial.

Now try 2 and -2 for x:

16∓48+16±48-32=0 (for 2 and -2).

2 | 1 -6  4 24  -32

     1  2 -8  -8 | 32

     1 -4 -4 16 |   0 = x³-4x²-4x+16

-2 | 1 -6   4  24  -32

      1 -2 16 -40 | 32

      1 -8 20 -16 |   0 = x³-8x²+20x-16.

So we have two lower degree polynomials with rational zeroes: 2, 4, 8, 16.

So we can try ±2 again:

x³∓4x=8-8=0 or -8+8=0 in each case, and -4x²+16=0 so we reduce the polynomial further:

2 | 1 -4 -4    16

     1  2 -4 | -16

     1 -2 -8 |    0 = x²-2x-8 = (x-4)(x+2).

Now we have all 4 zeroes: x=4, x=-2, x=2 (twice).

Polynomial is (x-4)(x+2)(x-2)².