The Appearance of a Polynomial Equation

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Polynomial equations sometimes come in disguise. For example, the formula:

y = (x +1) · (x - 4)² = (x +1) · (x - 4) · (x - 4)

does not look like a polynomial equation because it does not closely resemble the standard form of a polynomial equation given above.

However, if you FOIL this formula and carefully simplify then you can get the equation to resemble the standard form, and confirm that it is, indeed, a polynomial equation. Doing this:

y = (x +1) · (x - 4) · (x - 4)	(FOIL (x - 1) and (x - 4))
y = (x² - 3 · x - 4) · (x - 4)	(FOIL again)
y = x · (x² - 3 · x - 4) - 4 · (x² - 3 · x - 4)	(Multiply through)
y = x³ - 3 · x² - 4 · x - 4 · x² +12 · x +16	(Collect like terms)
y = x³ - 7 · x² + 8 · x +16	(Collect like terms)

This looks exactly like the standard form of the formula for a polynomial equation. So, although the equation did not initially look very much like a polynomial equation, it turned out to be a polynomial because it was possible to expand and simplify the equation, eventually making it resemble the standard form for a polynomial equation.