Theorems on Rationals

We present three theorems involving rational numbers.

nnth Power Theorem. Let n n be a positive integer. If a a is a positive integer such that a=rn a = r^n for some rational number r r, then r r must be an integer.

Proof: We apply the integer root theorem to the polynomial xna x^n - a . Since this polynomial has a rational root r r, this root must be an integer. _\square

Theorem 2. If the sum and product of two rational numbers are both integers, then the two rational numbers must be integers.

Proof: We have r1+r2=b r_1 + r_2 = b and r1r2=c r_1 \cdot r_2 = c, hence r1,r2 r_1, r_2 are rational roots of the polynomial x2bx+c x^2 - bx + c. By the integer root theorem, r1 r_1 and r2 r_2 are both integers. _\square

Theorem 3. A quadratic polynomial f(x)=ax2+bx+c f(x) = ax^2 + bx +c has integer coefficients. Then both of the roots are integers if and only if:

1. b24ac b^2 - 4ac is a perfect square,

2. a a divides both b b and c c.

Proof: By the Remainder Factor Theorem, if the roots are integers n n and m m, then f(x)=a(xn)(xm) f(x) = a (x-n)(x-m) . By comparing terms, we obtain b=a(n+m) b = -a(n+m) and c=anm c = anm. Hence, (2) is satisfied. Furthermore, b24ac=a2(n+m)24a×anm=a2(nm)2, b^2 - 4ac = a^2 (n+m)^2 - 4 a \times anm = a^2 (n-m)^2, so (1) is also satisfied.

Conversely, by the quadratic formula, the roots of f(x)f(x) are b±b24ac2a \frac {-b \pm \sqrt{b^2- 4ac} }{2a} . By condition (1), it follows that the roots are rational. By condition (2) and Vieta's formula, both the sum and product of the roots are integers. Hence by Theorem 2, the roots are integers. _\square

Note by Calvin Lin
7 years, 4 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link]( link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


Sort by:

Top Newest

what's &nbsp?

Daniel Lim - 6 years, 11 months ago

Log in to reply

Something like a non-breaking space.

Edit: Actually, it's  

Sean Ty - 6 years, 11 months ago

Log in to reply

It was meant to be a non-breaking space, to separate out indented paragraphs. I wanted the theorems to be displayed individually, instead of in a long chunk of text.

Calvin Lin Staff - 6 years, 11 months ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...