Question Number 3451 by prakash jain last updated on 13/Dec/15
![Consider a polynomial equation Σ_(i=0) ^n a_i x^i =0, a_i ∈Z Prove that if a+b(√c) is a root of the above equation then a−b(√c) is also a root. a,b,c∈Z, c is not a whole square.](https://www.tinkutara.com/question/Q3451.png)
$$\mathrm{Consider}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{equation} \\ $$$$\underset{{i}=\mathrm{0}} {\overset{{n}} {\sum}}{a}_{{i}} {x}^{{i}} =\mathrm{0},\:{a}_{{i}} \in\mathbb{Z} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}\:{a}+{b}\sqrt{{c}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{root}\:\mathrm{of}\:\mathrm{the}\:\mathrm{above} \\ $$$$\mathrm{equation}\:\mathrm{then}\:{a}−{b}\sqrt{{c}}\:\mathrm{is}\:\mathrm{also}\:\mathrm{a}\:\mathrm{root}. \\ $$$${a},{b},{c}\in\mathbb{Z},\:{c}\:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{whole}\:\mathrm{square}. \\ $$
Commented by prakash jain last updated on 13/Dec/15
![yes.](https://www.tinkutara.com/question/Q3457.png)
$${yes}. \\ $$
Commented by Filup last updated on 13/Dec/15
![ah, i see. You said c≠whole number if c∈Z, c is integer. you mean (√c) is not whole?](https://www.tinkutara.com/question/Q3456.png)
$${ah},\:\mathrm{i}\:\mathrm{see}.\:\mathrm{You}\:\mathrm{said}\:{c}\neq{whole}\:{number} \\ $$$$\mathrm{if}\:{c}\in\mathbb{Z},\:{c}\:{is}\:{integer}. \\ $$$$ \\ $$$$\mathrm{you}\:\mathrm{mean}\:\sqrt{{c}}\:\mathrm{is}\:\mathrm{not}\:\mathrm{whole}? \\ $$
Commented by Filup last updated on 13/Dec/15
![a, b∈Z c∈R](https://www.tinkutara.com/question/Q3454.png)
$${a},\:{b}\in\mathbb{Z} \\ $$$${c}\in\mathbb{R} \\ $$
Commented by prakash jain last updated on 13/Dec/15
![c∈Z, (√c) is a surd or complex.](https://www.tinkutara.com/question/Q3455.png)
$${c}\in\mathbb{Z},\:\sqrt{{c}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{surd}\:\mathrm{or}\:\mathrm{complex}. \\ $$