Question

Find the complete solution set. (Enter your answers as a comma-separated list.)
x^3 +8x^2+5x-14=0; {1}

Answer 1

P(x) = x³ + 8 x² + 5 x - 14

To find zeros for polynomials of degree 3 or higher it is used Rational root test.

The Rational root theorem tells you that if the polynomial has a rational zero then it must be a fraction ± p / q

Where p is a factor of the trailing constant and q is a factor of the leading coefficient.

The factor of the leading coefficient is 1 so q = 1

The factors of the trailing constan - 14 are 1 , 2 , 7 , 14

Then the Rational roots test yields the following possible solutions:

± p / q = ± p / 1 = ± p

x = ± 1 , x = ± 2 , x = ± 7 , x = ± 14

Substitute the possible roots one by one into the polynomial P(x) to find the actual roots.

P( - 1 ) = - 12

P( 1 ) = 0

P( - 2 ) = 0

P( 2 ) = 36

P( - 7 ) = 0

P( 7 ) = 756

P( - 14 ) = - 1260

P( 14 ) = 4368

The the solutions are:

x = - 7 , x = - 2 , x = 1

- 7 , - 2 , - 1