Obtain all zeros of the polynomial f(x) = 2x⁴ + x³ – 14x² – 19x – 6, if two of its zeros are -2 and -1.
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x = -1/2 and x = 3.
Step-by-step explanation:
Polynomial f(x) = 2x⁴ + x³ – 14x² – 19x – 6
–2 and -1 are zeroes of the polynomial.
Then (x+2) and (x+1) are factors of the polynomial and they divide the polynomial.
So (x+2)*(x+1) = x²+3x+2
2x⁴ + x³ – 14x² – 19x – 6 / (x²+3x+2) = 2x² -5x - 3
Now let's factorize 2x² -5x - 3
2x² -5x - 3 = 0
2x² -6x +x -3 = 0
2x (x-3) +1(x-3) = 0
(2x+1) (x-3) = 0
So the other roots are x = -1/2 and x = 3.
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