Let P(x) = 3x³ - 2x² + 5x + 4. Does the following equation have a root in the interval [-1, 0]? 3x³ - 2x² + 5x + 4 = 0

Question

Let P(x) = 3x³ - 2x² + 5x + 4.

Does the following equation have a root in the interval [-1, 0]?

3x³ - 2x² + 5x + 4 = 0

Answer 1

P(x) = 3x³ - 2x² + 5x + 4

Since P(x) is a polynomial function, it is continuous for all x.

P(-1) = -6 < 0 and P(0) = 4 > 0

By the Intermediate Value Theorem, P(c) = 0 for at least one real number,c, between -1 and 0.