Prove: if&nbsp;px2-(p+q)x+p=0&nbsp;has an integer root,&nbsp;px2+qx+p2-q=0&nbsp;has an integer root too, for com'&nbsp;p,q∈N

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Prove: if&amp;nbsp;px2-(p+q)x+p=0&amp;nbsp;has an integer root,&amp;nbsp;px2+qx+p2-q=0&amp;nbsp;has an integer root too, for com&#039;&amp;nbsp;p,q∈N

Melody Gamble · Accepted Answer

Step 1In this answer, p and q are co&#039; integers such that&amp;nbsp;q≥0&amp;nbsp;(and p can be zero or negative). We want to show that if&amp;nbsp;px2−(p+q)x+p=0&amp;nbsp;has an integer solution, then so does&amp;nbsp;px2+qx+p2−q=0If&amp;nbsp;x=t&amp;nbsp;is an integer root of&amp;nbsp;px2−(p+q)x+p=0, thent(p+q−pt)=p.If&amp;nbsp;t=0, then&amp;nbsp;p=0. Because&amp;nbsp;gcd(p,q)=1, this means&amp;nbsp;q=1. Then, the equation&amp;nbsp;px2+qx+p2−q=0&amp;nbsp;which is equivalent to&amp;nbsp;x−1=0, has an integer solution&amp;nbsp;x=1From now on, suppose that&amp;nbsp;t≠0. That is,&amp;nbsp;p=st&amp;nbsp;for some integer s. Hence,p+q−pt=pt=s.Therefore,q=s+p(t−1)=s+st(t−1)=s(t2−t+1).Therefore, s divides p and q, which implies&amp;nbsp;s=±1&amp;nbsp;as p and q are co&#039;. Since&amp;nbsp;q≥0, we get&amp;nbsp;s=1Therefore,&amp;nbsp;p=t&amp;nbsp;and&amp;nbsp;q=t2−t+1. Hence,px2+qx+p2−q=tx2+(t2−t+1)x+(t2−(t2−t+1))That is,&amp;nbsp;x=1−t&amp;nbsp;is an integer solution to the quation&amp;nbsp;px2+qx+p2−q=0The condition&amp;nbsp;q≥0&amp;nbsp;is necessary. Note that&amp;nbsp;p=−2&amp;nbsp;and&amp;nbsp;q=−3&amp;nbsp;satisfies the condition that&amp;nbsp;px2−(p+q)x+p=0&amp;nbsp;has an integer solution, but&amp;nbsp;px2+qx+p2−q=0&amp;nbsp;does not have an integer solution.

Jared Kemp · Accepted Answer

From the first equation, the product of the roots is 1. So they are both 1 or both -1. If they are both 1, substituting x=1 into the first equation gives q=p. Since p and q are co-&#039;, p=1,q=1 or p=−1,q=−1 If they are both -1, substituting x=−1 into the first equatio gives q=−3p. Since p and q are co-&#039;, p=1,q=−3 or p=−1,q=3 In all four cases for p and q, substituting the values for p and q into the second equation gives solutions for x that are integers.

Prove: if \(\displaystyle{p}{x}^{{2}}-{\left({p}+{q}\right)}{x}+{p}={0}{Z}{K}{h}{a}{s}{a}{n}\int{e}\ge{r}{\sqrt[,]{{P}}}{S}{K}{p}{x}^{{2}}+{q}{x}+{p}^{{2}}-{q}={0}\) has an integer root

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New Questions in Algebra I