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The Appearance of a Polynomial Equation
The Appearance of a Polynomial Equation
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))
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| 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.
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