Prove an expression is rational

In summary, the given rational numbers satisfy a specific equality and the expression $\sqrt{(c-3)(c+1)}$ can be proven to be rational by manipulating the given equality. The final result is that $\sqrt{(c-3)(c+1)}$ can be written as a rational expression.
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anemone
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The rational numbers $a,\,b,\,c$ (for which $a+bc$, $b+ac$ and $a+b$ are all non-zero) satisfy the equality $\dfrac{1}{a+bc} =\dfrac{1}{a+b}-\dfrac{1}{b+ac}$.

Prove that $\sqrt{(c−3)(c+1)}$ is rational.
 
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anemone said:
The rational numbers $a,\,b,\,c$ (for which $a+bc$, $b+ac$ and $a+b$ are all non-zero) satisfy the equality $\dfrac{1}{a+bc} =\dfrac{1}{a+b}-\dfrac{1}{b+ac}$.

Prove that $\sqrt{(c−3)(c+1)}$ is rational.
[sp]Write it as $\dfrac{1}{a+b} = \dfrac{1}{a+bc} + \dfrac{1}{b+ac} = \dfrac{(a+b)(1+c)}{(a+bc)(b+ac)}$. Then $$(a+b)^2(1+c) = (a+bc)(b+ac) = ab(1+c^2) + (a^2+b^2)c,$$ $$(a+b)^2 + 2abc = ab(1+c^2),$$ $$(a+b)^2 = ab(c-1)^2.$$ This shows that $ab = \Bigl(\dfrac{a+b}{c-1}\Bigr)^2$. In other words, $\sqrt{ab} = \Bigl|\dfrac{a+b}{c-1}\Bigr|,$ which is rational.

Also, $c-1 = \dfrac{\pm(a+b)}{\sqrt{ab}}$, so that $c-3 = \dfrac{\pm(a+b)}{\sqrt{ab}} - 2 = \dfrac{\pm(a+b) - 2\sqrt{ab}}{\sqrt{ab}}$, and $c+1 = \dfrac{\pm(a+b)}{\sqrt{ab}} + 2 = \dfrac{\pm(a+b) + 2\sqrt{ab}}{\sqrt{ab}}.$

Therefore $(c-3)(c+1) = \dfrac{(a+b)^2 - 4ab}{ab} = \dfrac{(a-b)^2}{ab}$, from which $\sqrt{(c-3)(c+1)} = \dfrac{|\,a-b\,|}{\sqrt{ab}}$, which is rational.[/sp]
 

FAQ: Prove an expression is rational

How do you prove an expression is rational?

One way to prove an expression is rational is to show that it can be written as a ratio of two integers, where the denominator is not equal to zero.

Can you use the rational root theorem to prove an expression is rational?

Yes, the rational root theorem states that if a polynomial has rational roots, they must be in the form of p/q where p is a factor of the constant term and q is a factor of the leading coefficient.

Are all rational expressions also polynomial expressions?

Yes, all rational expressions can be written as a polynomial expression with a finite number of terms.

Is it possible for an expression to be both irrational and rational?

No, an expression cannot be both irrational and rational. It can only be one or the other.

Can you prove an expression is rational using the fundamental theorem of algebra?

Yes, the fundamental theorem of algebra states that any polynomial expression with complex coefficients can be factored into linear or quadratic factors, which can then be written as rational expressions.

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