Question

if two zeros of the polynomial x⁴-13x³-33x²+39x+90 are root 3 and minus root 3 find the other zeros of the polynomial​

Answer 1

Answer:

Given roots of polynomial are √3 & –√3

then (x–√3) (x+√3) = x²–3 is factor of polynomial.

i.e

${x}^{4} - 13 {x}^{3} - 33 {x}^{2} + 39x + 90 = 0 \\ {x}^{4} - 13 {x}^{3} - 30 {x}^{2} - 3 {x}^{2} + 39x + 90 = 0 \\ {x}^{2} ( {x}^{2} - 13x - 30) - 3( {x}^{2} - 13x - 30) = 0 \\ ( {x}^{2} - 13x - 30)( {x}^{2} - 3) = 0 \\ i.e \: {x}^{2} - 13x - 30 = 0 \\ {x}^{2} - 15x + 2x - 30 = 0 \\ x(x - 15) + 2(x - 15) = 0 \\ (x - 15)(x + 2) = 0 \\ so \: \: x = 15 \: \: and \: \: - 2$

so other roots are 15 & –2.

Ans.