Square Roots of Polynomials

Algebra Level 5

When we have a equation involving square roots of polynomials, we typically rearrange square root terms, and square both sides to get rid of the square roots. As an example: $\begin{aligned} \sqrt{x} +\sqrt{x+1}& = \sqrt{x+2} \\ 2x+1+2\sqrt{x^2+x}& =x+2 \\ 2\sqrt{x^2+x}& =-x+1 \\ 4x^2+4x&=x^2-2x+1 \\ 3x^2+6x-1& = 0 \end{aligned}$ As shown in this example, we are guaranteed to be able to convert three square roots of first degree polynomials into a second degree polynomial by rearranging terms and squaring both sides multiple times. Consider equations of the form $\sqrt{P_1}+\sqrt{P_2}+...\sqrt{P_m}=\sqrt{P_{m+1}}+...\sqrt{P_n}$ where each $P$ stands for some first degree polynomial. Let $f(n)$ be the degree of the least degree polynomial we are guaranteed to be able to obtain (by the process of rearranging and squaring both sides) from equations of the given form. What is the maximum possible value of $f(n)$ ?

6 8 2 7 5

f(n)

can be arbitrarily large 4 3

1 solution

John Ross
Jun 27, 2018

We can only reduce equations with four terms or less into polynomials (where a term is a square root of a polynomial or a polynomial). If we square one side with $a$ terms (one or zero of which is a polynomial), we will get $\binom{a}{2}$ square root terms plus one polynomial term. This is because a square root polynomial times another square root polynomial or just a polynomial will give you another square root polynomial. This means that the number of new square root terms will be the number of ways to pair up $a$ old square root terms. The extra polynomial will come from terms times themselves. Thus if an equation has $a$ terms on one side and $b$ terms on the other, squaring both sides will give you $\binom{a}{2}+\binom{b}{2}+1$ new terms. $\binom{a}{2}$ grows more quickly than $a$ and the two are equal at $a=3$ . This means that if both sides have three terms ( $n=6$ ), squaring both sides will actually increase the number of square root terms. Splitting the sides unevenly doesn't help either because the most square root terms one could lose by squaring would be 1, and $\binom{a}{2}-a>1$ for $a \geq 4$ . It turns out $n=5$ doesn't work either because each time we square the equation, we still have 5 terms. Therefore we can only reduce equations with four terms or less into polynomials. An equation with four first-degree terms will reduce into a degree $\boxed{4}$ polynomial

Square Roots of Polynomials

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