(1+x)n=1+nx+ n(n-1)2x2+. s+xn n in Z_>0 Expansion in ascending powers (1+3x)5=1+ 51(3x)+ 52(3x)2+ 53(3x)3+. s =1+15x+90x2+270x3+. s Coefficient in a product (2-x)(1+3x)5=(2-x)(1+15x+90x2+. s) Coefficient of x2: 2(90)+(-1)(15)=165 To find a coefficient in a product, identify every pair of powers whose exponents add to the target. For x2, the constant times the quadratic term and the linear times the linear term both contribute.

\((1+x)^n=1+nx+\dfrac{n(n-1)}2x^2+\cdots+x^n\qquad n\in\mathbb Z_{>0}\)

\(\begin{aligned} (1+3x)^5&=1+\dbinom{5}{1}(3x)+\dbinom{5}{2}(3x)^2+\dbinom{5}{3}(3x)^3+\cdots \\ &=1+15x+90x^2+270x^3+\cdots \end{aligned}\)

\((2-x)(1+3x)^5=(2-x)(1+15x+90x^2+\cdots)\)

Coefficient of \(x^2\): \(2(90)+(-1)(15)=165\)

To find a coefficient in a product, identify every pair of powers whose exponents add to the target. For \(x^2\), the constant times the quadratic term and the linear times the linear term both contribute.

First terms and product coefficients

First terms and product coefficients