How to find the error

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HOW TO FIND AN ERROR

You have multiplied two long polynomials. How to check if you made a mistake in your calculations?

We present two methods that allow us to find some possibilities of error (but, of course, do not provide a complete guarantee of the correctness of the verified calculations).

Method 1. Take the sum of the coefficients of the first polynomial and multiply it by the sum of the coefficients of the second. Should get the sum of the coefficients of the product. This will be the value of the polynomial at $x=1$.

The above method allows you to find errors in arithmetic (for example, if the “equality” $2 \cdot 3=5$ was used), but not in algebra. Let’s say, if at some stage you wrote $2x^2 \cdot 3x^3=6x^4$, then this method will not produce an error.

Method 2. It differs from the first in that the coefficients for even powers $x$ are taken with a plane sign, and for odd ones with a minus sign. Otherwise, this method is the same as the first. Here we get the value of the polynomial at $x=-1$.

The second method is more complicated than the first; this also allows us to identify some algebraic errors. When checking for “equality” $2x^2 \cdot 3x^3=6x^4$ the result is $2 \cdot (-3)=6$. This equality is absurd, which means there was an error in the calculations.