How many positive and negative real zeros can f(x) = x7 − 4x5 − x2 + 4 have?
Hint
f(-x) = -x7 + 4x5 – x2 + 4
Answer
It can have 2 or 0 positive real roots and 3, 1, or 0 negative real roots.
How many positive and negative real zeros can f(x) = x9 + 4x3 − x2 + 2x + 6 have?
f(-x) = -x9 – 4x3 – x2 – 2x + 6
It can have 2 or 0 positive real roots and 1 or 0 negative real roots.
How many positive and negative real zeros can f(x) = -x7+ 4x4− x3 + x2 + 5x – 5 have?
f(-x) = x7+ 4x4 + x3 + x2 – 5x – 5
It can have 4, 2, or 0 positive real roots and 1 or 0 negative real roots.
Use synthetic division to check if -3 is an upper bound of y = 4x7 + 8x4 − 12x3 + 4x2 + 2x – 1.
Check the sign of a before starting the problem.
We cannot use our method to check upper bounds when a is negative. So, we boant know.
Use synthetic division to check if -3 is a lower bound and 3 an upper bound of y = 4x7 + 8x4 − 12x3 + 4x2 + 2x – 1.
The signs of the quotient are all positive when using synthetic division on a = 3, and the signs alternate when synthetically dividing a = -3.
Yes to both.
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