Question

Please explain this problem and How to solve this problem please explain step by step. Thanks

Determine if the lower bound theorem identifies -7 as a lower bound for the real zeros of
f (x) = 5x3 - 2x2 - 5x - 2
A Yes
B NO

Answer 1

Answer: A) Yes

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Step-by-step explanation:

Let c < 0 be a test root of f(x). The key here is that c must be negative.

If we use this value of c in the synthetic division table, and we get the third row alternating in signs, then we have proven c is a lower bound of the roots of f(x). This is what the lower bound theorem tells us.

In this case, c = -7 fits the description of c < 0.

Refer to the diagram below to see the synthetic division. The third row of values is: 5, -37, 254, and -1780

The terms alternate in sign which is sufficient evidence to conclude -7 is a lower bound for the real zeros of f(x).

What this means is that we don't need to look lower than -7 since we know -7 is like a "floor" of sorts. It's not the highest lower bound as there may be higher floor values.

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For more information, search out
`\text{``lower bound theorem'' algebra}` where the quotes are needed. One of the results is from the West Texas A&M University (WTAMU) Virtual Math Lab, which may be handy.