Question

Obtain all the zeros of 2x⁴+5x³-8x²-17x-6 if three of its zeros are -1, -3, 2.​

Answer 1

The above quartic equation can be factorised into linear factors using the rational zeroes theorem or directly taking the given three roots as the factors one by one or even simultaneously.

we have, x = –1 , –3 , 2

so, a cubic equation which is a factor of the given quartic is :

(x+1)(x+3)(x–2) = x³ + 2x² –5x –6

when we perform polynomial division we get, the other factor as

$\frac{ 2{x}^{4} + 5 {x}^{3} - 8 {x}^{2} - 17 {x} - 6}{ {x}^{3} + 2 {x}^{2} - 5x - 6 } = (x + \frac{1}{2} )$

So, the fourth root is –0.5