Notes
Before you start: find a binomial coefficient and multiply polynomials.
If \(A(x)=\sum a_i x^i\) and \(B(x)=\sum b_j x^j\) are polynomials in nonnegative powers, the coefficient of xᵐ in A(x)B(x) is \(\sum_{i=0}^m a_i b_{m-i}\). List all power pairs that add to m. For m=2 these are (0,2), (1,1) and (2,0); the middle product is easy to miss.
For ordinary polynomials, powers higher than m cannot contribute to xᵐ, so it is safe to truncate each factor above that degree. This shortcut needs care with negative powers: a high positive power can combine with a negative one to give the target.
Before multiplying, look for \((1+x)^n(1-x)^n=(1-x^2)^n\). This contains only even powers. More generally, if F is a polynomial, its even- and odd-power coefficient sums are \((F(1)+F(-1))/2\) and \((F(1)-F(-1))/2\). This follows because (−1) to an even power is 1 and to an odd power is −1.
Recall later
The coefficient of x² in \((1+2x)(1+3x)\) is 6, from the pair (1,1). Adding the separate x² coefficients would give the wrong answer.
Common mistakes
- Missing the cross-product when collecting a coefficient.
- Keeping only same-degree terms instead of pairs whose powers add to the target.
- Applying polynomial truncation to factors with negative powers without checking other possible pairs.