Both algebraic expressions and polynomials are a combination of terms grouped together. What is the difference between them?
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asked Dec 18, 2014 at 18:27
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2 Answers
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"Polynomial" is a precisely defined term. A polynomial is constructed from constants and variables by adding and multiplying. One could add "subtracting", but $x-y$ is $x+(-1)y$, so adding and multiplying are enough.
"Algebraic expression" is not a precisely defined term. Algebraic expressions include many things that are not polynomials, including rational funtions, which come from dividing polynomials, and things like $\sqrt{x}$.
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1$\begingroup$ So the difference is that the polynomials can only include +/-/÷/x, variables, coefficients and constants, while algebraic expressions can include anything, for e.g. square roots? $\endgroup$ Dec 18, 2014 at 18:51
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1$\begingroup$ @Artemisveras : Actually one should not allow division. The set of polynomials is closed under addition, subtraction, and multiplication, but not division. ${}\qquad{}$ $\endgroup$ Feb 20, 2015 at 6:00
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$\begingroup$ So if there's division in an expression, it's not a polynomial? $\endgroup$ Feb 20, 2015 at 13:34
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2$\begingroup$ @Artemisveras : One can divide by scalars, since one just gets another scalar, and those are allowed. But dividing by anything containing variables will get you something that is not a polynomial, unless that factor in the denominator cancels out because the same factor appears in the numerator. ${}\qquad{}$ $\endgroup$ Feb 20, 2015 at 18:52
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Some difference is:
Algebraic Expression: may not be a continuous function on $R = (-\infty,\infty)$, but Polynomial is. Example: $\dfrac{x}{x+1}$, and $x^2-1$. The former is not defined at $x=-1$, while the latter is continous throughout $R$.
