Question

If p(x) is a polynomial have sum of zeroes is 0 and product is -6 .find all zeroes when p(x) is x⁴+x³-12x²-6x+36

Answer 1

We've to find the first zero by checking with different numbers

when x=1,
=>1+1-12-6+36=20 which is not equal to zero

now try with x=2,
=>16+8-48-12+36=0so x=2 is a zero of the polynomial

since (x-2) is a factor of the polynomial divide p(x) with (x-2)

        
        x³ + 3x² - 6x -18
      -------------------------
x-2║x⁴ + x³-12x²-6x+36
        x⁴ -2x³
       -     +
      -------------------------
            3x³-12x²
            3x³- 6x²
           -     +
      -------------------------
                 -6x²-6x
                 -6x²+12x
                 +    -
     --------------------------
                        -18x+36
                        -18x+36
                        +     -
     ---------------------------
                          0

     =>(x³+3x²-6x-18) (x-2) =  (x⁴+x³-12x²-6x+36)

   Now we can easily find zeroes from (x³+3x²-6x-18)
 =>(x³+3x²-6x-18) = x²(x+3)-6(x+3) 
    = (x²-6)(x+3)=0
     Since (x+3) is factor x=-3 is zero the polynomial

Now factorize x²-6
x²-6=0
x²=6
x=+or- √6
x= √6 ,-√6

Therefore the zeroes are √6,-√6,2,-3