Question

Find the rational roots of x^4+5x^3+7x^2−3x−10=0

Answer 1

To find the rational roots of the given polynomial equation, we can use the rational root theorem.

The rational root theorem states that if a rational number r is a root of a polynomial equation with integer coefficients, then r must be a factor of the constant term divided by a factor of the leading coefficient.

In this case, the constant term is -10 and the leading coefficient is 1. The factors of -10 are ±1, ±2, ±5, and ±10, and the factors of 1 are ±1.

Therefore, the possible rational roots of the equation are ±1, ±2, ±5, and ±10.

We can now check each of these values by substituting them into the equation to see if they make it true.

If we substitute x = 1 into the equation:

(1)^4 + 5(1)^3 + 7(1)^2 − 3(1) − 10 = 1 + 5 + 7 − 3 − 10 = 0

Therefore, x = 1 is a root of the equation.

If we substitute x = -1 into the equation:

(-1)^4 + 5(-1)^3 + 7(-1)^2 − 3(-1) − 10 = 1 - 5 + 7 + 3 − 10 = -4 + 10 = 6

Therefore, x = -1 is not a root of the equation.

Similarly, by checking the remaining possible rational roots, we find that x = -2 is a root of the equation.

Therefore, the rational roots of the given equation are x = 1 and x = -2.