Question

find all the zeroes of the polynomial 2x^4+5x^3-11x^2-20x+12 if two of its zeroes are 2 and -2

Answer 1

Since two zeroes are 2 and -2, (x-2)(x+2)=x^2-4 must be a factor. Algebraic long division gives us:

            2x^2+5x-3 = (2x-1)(x+3) so the other zeroes are 1/2 and -3.

x^2-4 ) 2x^4+5x^3-11x^2-20x+12

            2x^4           -8x^2

                     5x^3  -3x^2-20x

                     5x^3           -20x

                               -3x^2       +12

                               -3x^2       +12

                                        0

 

                              

Answer 2

We know if x=a is a zero of a polynomial and then x-a is a factor of f(x). Since 2 and -2 are zeros of f(x) Therefore: (x-2) (x+2) = x²-4 (x²-4) is a factor of f(x) Now, we divide 2x⁴-5x³-11x²+20x+12 by g(x)=(x²-4) to find the zero f(x). By using division algorithm we have, f(x)=g(x)×q(x)-r(x) 2x⁴-5x³-11x²+20x+12=(x²-4)(2x²-5x-3) = (x-2)(x+2)[2x(x-3)+1(x-3)] =(x-2)(x+2)(x-3)(2x+1) Hence the zeros of the polynomial are; 2,-2,3,-1/2