MATHEMATICS: Rational Zero Theorem?

Question

Find the rational zeros of the polynomial function if they exist. If possible, find the other zeros. Then write the equation in factored form.

A) P(x) = x^3 - x^2 - 4x - 2

B) P(x) = 2x^3 -3x^2 -7x - 6



Please show short solutions. Thank you so much in advance!

Answer 1

This is the theorem you need: If the polynomial equation (a_n)*x^n + (a_(n-1))*x^(n-1) + ... + (a_1)*x + a_0 = 0 has a rational root a/b, then a divides a_0 and b divides a_n.

For example, in part A, if a/b is a rational root, then a divides -2 and b divides 1. Therefore, a must be 1, -1, 2, or -2 and b is 1 or -1. Consequently, the possibilities for rational roots are 1, -1, 2, -2. Test each one to find which work.

Answer 2

A) P(x) = x^3 - x^2 - 4x - 2
P(-1) = -1 - 1 + 4 - 2 = 0
x+1 is a factor
Divide (x^3 - x^2 - 4x - 2) by (x+1)
x^3 - x^2 - 4x - 2 = (x+1)(x^2 - 2x - 2)
Rational roots: {-1}
Irrational roots: [2±srqt(4+8)] / 2 = 1±sqrt(2.5)


B) P(x) = 2x^3 - 3x^2 - 7x - 6
P(3) = 54 - 27 + 21 - 6 = 0
x+3 is a factor
Divide (x^3 - x^2 - 4x - 2) by (x+3)
2x^3 - 3x^2 - 7x - 6 = (x-3)(2x^2 + 3x + 2)
Rational zeros: {-1}
This expression has no other real zeros