Question

Find zeroes of polynomial 2x 4 +7x ​3 -19x2-14x+30 given two of its zeroes are root2 and -root2

Answer 1

Zeroes of polynomial are √2 and -√2
Factor will be 
(x-√2)(x+√2)
=(x²-2)

now, dividing the polynomial with this factor,

x²-2)2x⁴ +7x³ -19x² -14x +30(2x²+7x-15
       2x⁴         -4x²
      -             +
       0     7x³-15x²-14x+30
              7x³        -14x
             -             +
              0    -15x  0   +30
                    -15x       +30
                    +            - 
                     0              0

So, the factors are (x²-2)(2x²+7x-15)
=(x²-2)(2x²+10x -3x -15)
=(x²-2)[2x(x+5)-3(x+5)]
=(x²-2)(x+5)(2x-3)

so, other zeroes are -5 and 3/2

Answer 2

Answer:

Zeroes of polynomial are √2 and -√2

Factor will be 

(x-√2)(x+√2)

=(x²-2)

now, dividing the polynomial with this factor,

x²-2)2x⁴ +7x³ -19x² -14x +30(2x²+7x-15

      2x⁴         -4x²

     -             +

      0     7x³-15x²-14x+30

             7x³        -14x

            -             +

             0    -15x  0   +30

                   -15x       +30

                   +            - 

                    0              0

So, the factors are (x²-2)(2x²+7x-15)

=(x²-2)(2x²+10x -3x -15)

=(x²-2)[2x(x+5)-3(x+5)]

=(x²-2)(x+5)(2x-3)

so, other zeroes are -5 and 3/2

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