Determine if the statement is True or False. Justify your response.?

Question

If the coefficients of a polynomial function are all positive, then the polynomial has no positive zeros.

Answer 1

True.

Let's prove this by contradiction. Assume that all the coefficients are positive and you have at least one positive root r1.

r1 > 0

So the polynomial would be:
f(x) = (x - r1)(x - r2)

Multiplying out:
x² - r1x - r2x + r1r2
x² + (-r1 - r2)x + r1r2

Okay, we are good for the coefficient on x², it is 1.
If r1 is positive, then r2 must also be positive to make the final coefficient r1r2 positive.

And if r1 and r2 are both positive then (-r1 - r2) will be negative. But this violates our initial condition, so there is no way to have all the coefficients be positive and have a positive roots.

Therefore the statement is true:
If the coefficients of a polynomial function are all positive, then the polynomial has *no* positive zeros.

Answer 2

Exactly

How can you get a root is all the involved numbers are positive ones?

Example:

4x + 1

Which value of x, x positive, would make this polynomials 0?

Ana

Answer 3

true
P(x) = a_n x^n +... + a_1 x + a_0
if x >0 then P(x)> a_0 > 0