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
How do I figure this out?
You could sketch the curve and see
or
take a look at this video from Khan Academy, one of the best sites on line your can find.
Sal explains the Fundamental Theorem of Algebra
https://www.khanacademy.org/math/algebra2/polynomial-functions/fundamental-theorem-of-algebra/v/fundamental-theorem-of-algebra-intro
To determine the number and type of roots of the given equation, you can use the concept of Descartes' Rule of Signs. Here's how you can figure it out step by step:
1. Count the number of sign changes in the coefficients of the equation:
- In the given equation, x^3 - 4x^2 + 50x + 7 = 0, there is only one sign change (from positive to negative) from the term -4x^2 to 50x.
2. Now, calculate the sign change for the equation f(-x):
- Replace x with -x in the equation, which gives -x^3 - 4x^2 - 50x + 7 = 0.
- Count the sign changes in this new equation: There are two sign changes (from positive to negative) from -x^3 to -4x^2, and from -4x^2 to -50x.
Based on Descartes' Rule of Signs:
- The number of positive roots of the equation is equal to the number of sign changes in the original equation or less than that by an even number. In this case, the number of positive roots is 1 or less than that by an even number (0).
- The number of negative roots of the equation is equal to the number of sign changes in the equation f(-x) or less than that by an even number. In this case, the number of negative roots is 2.
Therefore, the answer to the question "Which describes the number and type of roots of the equation x^3 - 4x^2 + 50x + 7 = 0?" is option B: 2 positive, 1 negative.