Find ALL the zeros real and complex of Pxx43x3x27x30Solution

Find ALL the zeros, real and complex of P(x)=x^4-3x^3+x^2+7x-30.

Solution

P(x)=x^4-3x^3+x^2+7x-30

Rational root theorem :

The factor of the leading coefficient (1) is 1 .The factors of the constant term (-30) are 1 2 3 5 6 10 15 30 . Then the Rational Roots Tests yields the following possible solutions:

±1/1, ±2/1, ±3/1, ±5/1, ±6/1, ±10/1, ±15/1, ±30/1

checking for all these points:

P(-2) =0

Divide :(x^4-3x^3+x^2+7x-30)/(x+2) = x^35x^2+11x15

again use rational root theorem:

The factor of the leading coefficient (1) is 1 .The factors of the constant term (-15) are 1 3 5 15 . Then the Rational Roots Tests yields the following possible solutions:

±1/1, ±3/1, ±5/1, ±15/1

P(3) =0

So, divide x^35x^2+11x15/( x-3) = x^2 -2x +5

solve the quadratic to get the roots:x = 1 +/-2i

So, zeros of polynomial are : x = -2 , 3 , 1 +/- 2i