Evaluate p(x) = 7x5 - 9x4 - x2 at 1 and 2 to determine if the Intermediate Value Theorem guarantees that a zero exists between the two values. a) p(1) = b) p(2)

Question

Evaluate p(x) = 7x⁵ - 9x⁴ - x² at 1 and 2 to determine if the Intermediate Value Theorem guarantees that a zero exists between the two values.

a) p(1) =    

b) p(2) 

Answer 1

p(x) = 7x⁵ - 9x⁴ - x² = x²(7x³ - 9x² - 1)

a) p(1) = (1²)·(7·1³ - 9·1² - 1) = 1·(7 - 9 - 1) = -3

b) p(2) = you do this one

The intermediate value theorem says that if one of the values (p(1) or p(2)) is positive and other is negative, then p(x) must = 0 somewhere in between (guaranteed). Otherwise, we don't know (not guaranteed).