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Q: There are some instances where it is better to factor a polynomial without first putting it in standard form One example is a quadratic equation that in nonstandard form contains a perfect tr?

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Actually, it is easier to define a polynomial as a function of a variable, x, in the form:p(x) = anxn + an-1xn-1 + ... + a2x2 + a1x + a0 where n is an integer and the ai are constants. A non-polynomial, then, is any function of x which contains one or more terms in x which are not in the form akxk.

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Here is an intuitive method for working this polynomial problem, which can be solved as easily as adding apples to apples and oranges to oranges. Think of the same-named variables and variable pairs as different "things." In this example, each polynomial contains none or one or more of these three different things: "ax", "by", and "c". So all we need to do is add (or subtract) up the "ax"s, then the "by"s, and then the "c"s and write it down. ax: 1 in the first polynomial + 2 in the second + none (0) in the third = 3 "ax"s. by: 1 in the first polynomial - 3 in the second +1 in the third = -1 "by"s. c: 1 in the first polynomial + 1 in the second - 1 in the third = 1 "c". The answer is the polynomial made up of the three results above: 3ax - 1by + 1c = 3ax - by + c

It depends on what it contains: none, if it is empty!It depends on what it contains: none, if it is empty!It depends on what it contains: none, if it is empty!It depends on what it contains: none, if it is empty!

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Actually, it is easier to define a polynomial as a function of a variable, x, in the form:p(x) = anxn + an-1xn-1 + ... + a2x2 + a1x + a0 where n is an integer and the ai are constants. A non-polynomial, then, is any function of x which contains one or more terms in x which are not in the form akxk.

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