Find the nature of the roots of the equation x^4+15x^2+7x-11=0

Solution

Descartes' Law of signs says there is one negative, one positive, and two complex roots.

find the nature of the roots of the equation (use descartes rule ) x⁴+15x²+7x-11=0

Well, let's break it down! This equation seems like it needs a little fun injected into it. Now, the nature of the roots depends on the discriminant, which is the part under the square root in the quadratic formula. If the discriminant is positive, we get two real and distinct roots. If it's zero, we get two real and identical roots. And if it's negative, well, we end up with a complex dance party with two complex roots!

Now, the discriminant for this equation is one mysterious creature: √(15^2 - 4 * 1 * (-11)). Let's do some math and find out what it's hiding. Oh, it seems to be approximately equal to 229. So, this means we have two real and distinct roots, ready to do their own thing!

To determine the nature of the roots of the given equation x^4 + 15x^2 + 7x - 11 = 0, you can use either the discriminant or apply Descartes' rule of signs.

Let's first identify the coefficients of the equation:
a = 1 (coefficient of x^4)
b = 15 (coefficient of x^2)
c = 7 (coefficient of x)
d = -11 (constant term)

Using the discriminant, we can find the nature of the roots. The discriminant (D) is calculated as D = b^2 - 4ac.

Substituting the values, we have:
D = (15)^2 - 4(1)(7)
D = 225 - 28
D = 197

Since the discriminant D is positive (D > 0), it means that there are two real roots.

Now, let's apply Descartes' rule of signs to determine the number of positive and negative real roots:

1. Count the number of sign changes when the equation is written in standard form:
From +15x^2 to +7x, there is one sign change.

2. Count the number of sign changes when you substitute -x for x in the equation:
From -15x^2 to +7x, there is one sign change.

Using this rule, we conclude that there is exactly one positive real root.

Since the equation has two real roots and only one positive real root, it must have one negative real root as well.

Therefore, the nature of the roots of the equation x^4 + 15x^2 + 7x - 11 = 0 is two real roots, one positive and one negative.