Which describes the number and type of roots of the equation x^3 - 4x^2 + 50x + 7 = 0?

a. 1 positive, 2 negative
b. 2 positive, 1 negative
c. 3 negative
d. 3 positive

I know I asked this before but I'm going over it and I don't understand it. I got help from someone and they said the answer was b, but I still don't understand it.

Using Descartes' Rule of Signs,

f(x) has 2 sign changes. so, 2+ or 0+
f(-x) has 1 sign change, so 1-

Looks like 2+, 1-

0+ and 1- is also a possibility, but not one of the choices...

(x-a)(x-b)(x-c) = 0

well -a * -b * - c = +7
they can not all be + or no +7 OUT
so they could all be - but then I would not get -4 anywhere OUT
if a is - and 2 others + that would get +7 YES
if a is + and two others - that gets for -7 OUT

To determine the number and type of roots of the equation x^3 - 4x^2 + 50x + 7 = 0, we can use Descartes' Rule of Signs.

Step 1: Count the number of sign changes in the equation.

In this case, there are two sign changes: from positive to negative when moving from -4x^2 to +50x, and from negative to positive when moving from +50x to +7.

Step 2: Determine the possible number of positive roots.

The possible number of positive roots is either equal to the number of sign changes or less than it by an even integer. In this case, there are two sign changes, so the possible number of positive roots is either 2 or 0.

Step 3: Substitute (-x) for x in the equation and repeat steps 1 and 2.

Substituting (-x) for x in the equation gives (-x)^3 - 4(-x)^2 + 50(-x) + 7 = -x^3 - 4x^2 - 50x + 7.

There are no sign changes in the new equation, which means there are no positive roots.

Step 4: Determine the possible number of negative roots.

The possible number of negative roots is either equal to the number of sign changes or less than it by an even integer. Since there are no sign changes in this case, the possible number of negative roots is 0.

Based on these steps, we can conclude that the equation x^3 - 4x^2 + 50x + 7 = 0 has 2 positive roots and 0 negative roots. Therefore, the correct answer is b.

To determine the number and type of roots of the equation x^3 - 4x^2 + 50x + 7 = 0, you can use the Fundamental Theorem of Algebra along with some algebraic methods. Here's how you can approach it:

Step 1: Identify the degree of the equation.
The equation x^3 - 4x^2 + 50x + 7 = 0 is a polynomial equation of degree 3 because the highest exponent in the equation is 3.

Step 2: Apply the Fundamental Theorem of Algebra.
The Fundamental Theorem of Algebra states that a polynomial equation of degree n has exactly n complex roots, including repeated roots.

Step 3: Look for rational roots.
To determine the rational roots (which are also the real roots) of the equation, you can apply the Rational Root Theorem. According to the Rational Root Theorem, the rational roots (if they exist) must be of the form p/q, where p is a factor of the constant term (in this case, 7) and q is a factor of the leading coefficient (in this case, 1).

In this case, the factors of 7 are ±1 and ±7, and the factors of 1 are ±1. So, the possible rational roots are ±1, ±7.

Step 4: Apply synthetic division to test the possible rational roots.
Using synthetic division, we can test each of the possible rational roots in the equation x^3 - 4x^2 + 50x + 7 = 0. If a tested root results in a remainder of 0, it means that the tested root is a solution of the equation.

By dividing x^3 - 4x^2 + 50x + 7 = 0 by each of the possible rational roots, you will find that none of them produces a remainder of 0. This implies that the equation does not have any rational roots.

Step 5: Use Descartes' Rule of Signs.
Descartes' Rule of Signs can help determine the number of positive and negative roots of a polynomial equation. Counting the sign changes in the coefficients of the terms in descending order, you can draw conclusions about the number of positive and negative roots.

For the equation x^3 - 4x^2 + 50x + 7 = 0, there are two sign changes. According to Descartes' Rule of Signs, this means the equation could have either two positive roots or no positive roots.

Step 6: Apply the Conjugate Root Theorem.
Since the equation x^3 - 4x^2 + 50x + 7 = 0 has real coefficients, the Conjugate Root Theorem states that if a + bi is a root of the equation (where a and b are real numbers), then its conjugate a - bi must also be a root.

Since the equation did not yield any rational roots, it can be concluded that both complex roots must be in conjugate pairs.

Putting it all together, we can determine that the equation x^3 - 4x^2 + 50x + 7 = 0 has:
- Two complex roots, which are a conjugate pair (no real solutions).
- No positive roots.

Therefore, the correct answer is c. 3 negative roots.