Question

Help quick! Medal for right answer.
Which statement about the polynomial function g(x) is true?

~ If all rational roots of g(x) = 0 are integers, the leading coefficient of g(x) must be 1.
~ If all roots of g(x) = 0 are integers, the leading coefficient of g(x) must be 1.
~ If the leading coefficient of g(x) is 1, all rational roots of g(x) = 0 must be integers.
~ If the leading coefficient of g(x) is 1, all roots of g(x) = 0 must be integers.

Answer 1

i think its A mmm (but not sure ")

Answer 2

The "Rational Root Theorem tells us that: Given a polynomial, \(a_0x^{n} + a_{1}x^{n-1} + ... + a_{n}\), all RATIONAL Roots must be of this form:

\(\dfrac{Factor\;of\;a_{n}}{Factor\;of\;a_{0}}\)

If '\(a_{0}\)' has factors other than 1, then Rational Roots COULD be non-integer.
If \(a_{0} = 1\), which implies no factor other than 1, all RATIONAL roots MUST be Integer. (Must have a denominator of 1)

Now, let's think about it...

"~ If all rational roots of g(x) = 0 are integers, the leading coefficient of g(x) must be 1."

This seems tempting, but we really don't know enough about the polynomial. What we REALLY don't know, is if there is a common numerical factor amongst the coefficients.

2x + 2 = 0, for example. Integer root (-1), but 2 is not 1. I really want to bite on this one, but it says only "polynomial", and not "polynomial with common numerical factors removed from the coefficients" or "polynomial with mutually prime coefficients" or something like that.

"~ If all roots of g(x) = 0 are integers, the leading coefficient of g(x) must be 1."

No good. Same reason as above.

"~ If the leading coefficient of g(x) is 1, all rational roots of g(x) = 0 must be integers."

Me likey!

"~ If the leading coefficient of g(x) is 1, all roots of g(x) = 0 must be integers."

No. It may not have any REAL roots at all.

For example, x^2 + 1 = 0

WONDERFUL Question. My compliments to the author!

Answer 3

Thank you so much for your help!

Answer 4

Now, you do one. Understand the problem and reason out the answer. Remember, it's not magic.

Answer 5

Do what exactly? List example equations?

Answer 6

Was this the only problem you had on your assignment? I was hoping for another. Oh, well.

Answer 7

No, sorry. It was just this question, but marvelous answer though. I appreciate you taking the time to explain it so thoroughly.

Answer 8

What would an example problem look like?

Answer 9

Make them up!

Name three possible rational roots of the polynomial: \(p(x) = 3x^{2} - 5x + 12\)