How2Day: Alternatives to Factoring

In algebra, a "polynomial" is a mathematical expression---"x^2 + 2x - 5 = 0" for example---containing two or more terms, be they variables like "x" or "x^2" or real numbers. One common method to solve polynomials is "factoring," wherein you simplify a polynomial to a product of sums and/or differences, such as "(x - 2)(x - 3)" instead of "x^2 - 5x + 6." If you don't want to factor, there are several alternatives.

Perhaps the most common alternative to factoring, the quadratic formula is also a bit longer and arguably more complicated. You can calculate the solutions---for which values of "x" the expression equals zero---for any polynomial "ax^2 + bx - c = 0" using the formula "x = (-b +/- [sqrt{b^2 - 4ac}])/(2a)." For the polynomial x^2 - 7x + 6, for example, this would entail the following calculations: x = (-7 + sqrt[7^2 - 4*1*6])/(2*1) and x = (-7 - sqrt[7^2 - 4*1*6]/2*1), or (-7 +/- sqrt25)/2, or (-7 + 5)/2 and (-7 - 5)/2, or "-6" and "-2."

As the "solution" to a polynomial is nothing more than where it crosses the "x" axis---in other words, where "y" = 0---a logical way to find these solutions is graphing a polynomial on a coordinate plane. For some polynomials---namely those with whole-number answers---this method works fine, although the actual plotting and drawing can take time. If the polynomial has a decimal answer, however, extrapolating this answer from a graph might require estimation---in other words, compromised accuracy and precision.

The least efficient alternative to factoring, the "guessing" technique, usually involves the use of the factoring "form." This can, however, be problematic. Knowing, for example, that a binomial a^2 - b^2 is always a "difference of squares"---(a + b)(a - b)---it's tempting to assume the same simple truth exists about any polynomial ax^2 + bx + c. Where guessing becomes difficult and time-consuming is during the process of checking your answers---multiplying out your guesses to see if they're right. Often, a difference in sign---(x + 5)(x - 5) vs. (x - 5)(x - 5)---can cause a huge difference in value---"x^2 - 25" or "x^2 - 10x + 25," respectively.

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