SOLUTION: On which interval does the Intermediate Value Theorem guarantee that the polynomial function f(x) = x⁴ + 7x² − 9x − 1 has a root?

A polynomial f(x) has a root on the interval [a,b] if f(a) and f(b)
have different signs.

f(x) = x4 + 7x2 − 9x − 1

has a root on the interval [1,2] because the value at the left endpoint

f(1) = (1)4 + 7(1)2 − 9(1) − 1 = 1+7(1)-9-1 = -2

is negative and the value at the right endpoint

f(2) = (2)4 + 7(2)2 − 9(2) − 1 = 16+7(4)-18-1 = 25

is positive.  

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It also has a root on the interval [-1,0] because the value at the left endpoint

f(-1) = (-1)4 + 7(-1)2 − 9(-1) − 1 = 1+7(1)+9-1 = 16

is positive and the value at the right endpoint

f(0) = (0)4 + 7(0)2 − 9(0) − 1 = 0+7(0)+0-1 = -1

is negative.

Edwin