Restricting the Roots Examples

How many positive and negative real zeros can f(x) = -4x⁴+2x³ − 3x² + 5x − 2 have?

We need to use Descartes Rule of Signs to figure this problem out.

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For f(x), we have 4 changes in sign. That means we can have 4, 2, or 0 positive real roots.

f(-x) = -4(-x)⁴ + 2(-x)³ – 3(-x)² + 5(-x) – 2

= -4x⁴–2x³ − 3x² – 5x − 2

This is all negative, all the time. What a downer. It does mean that we have 0 negative real roots, guaranteed. That cuts out half of the possible roots we would need to check.

Is 20 an upper bound for the roots of y = x⁴− x³−1000x + 1000?

The possible rational roots of the polynomial are ±1000, ±500, ±250, ±200, ±125, ±100, ±50, ±40, ±25, ±20, ±10, ±8, ±5, ±4, ±2 and ±1. We don't want to alarm you, but that's a lot.

If 20 is an upper bound, there's no point in checking to see whether 25, 40, 50, 100, 125, 200, 250, 500, or 1000 are roots. The less time we spend digging in the dirt looking for roots, the better.

We'll use Synthetic Division with 20 and see what we get.

Hey, where did x² go? Well, as long as she's back by dinner, and we account for her in our synthetic division, it's okay. The third row is all positive, which means 20 is an upper bound.

Is -2 a lower bound for y = 4x⁵ +6x⁴ − 12x³ +5x² +2x – 13? What about -3?

Some people, less diligent than you are, might say, "Of course -2 isn't a lower bound. Otherwise, they wouldn't also ask about -3. I'm gonna skip ahead and just check -3, and then go outside and throw litter in the park."

Geez, not only are they skimping on their math, they're also kind of a jerk. Don't be jerk; check both bounds.

Uh, yeah, -2 doesn't cut it as a lower bound. The signs don't alternate all of the way through. Guess we still need to check -3.

Okay, here we go. Those signs bounce between positive and negative like a pinball caught on the bumpers. -3 totally is a lower bound.