p(x) = 36x4 − 48x3 − 8x2 + 16x + 4 find the zeros and multiplicity

Question

i dont understand

Answer 1

p(1)=36-48-8+16+4=0, so x=1 is a zero.

p(x)=4(9x⁴-12x³-2x²+4x+1)

Use synthetic division to reduce the quadratic:

1 | 9 -12  -2  4    1

     9    9  -3 -5 | -1

     9   -3  -5 -1 |   0 = 9x³-3x²-5x-1.

This cubic is zero when x=1 (9-3-5-1=0) so x=1 is a duplicate zero (x-1)².

So divide by the zero again:

1 ) 9 -3 -5   -1

     9   9  6  | 1

     9   6  1  | 0 = 9x²+6x+1 = (3x+1)², so we have a pair of duplicate zeroes: 1 and -⅓.

p(x)=4(x-1)²(3x+1)².