Question # write the polynomial P(x)=x^{2}, if possible as a linear combination of the polynomials 1+x,2+x^{2},−x.

Polynomials
ANSWERED write the polynomial $$\displaystyle{P}{\left({x}\right)}={x}^{{{2}}},$$ if possible as a linear combination of the polynomials $$\displaystyle{1}+{x},{2}+{x}^{{{2}}},−{x}.$$ 2021-01-14

Let $$\displaystyle{x}{2}={a}{\left({1}+{x}\right)}+{b}{\left({2}+{x}^{{{2}}}\right)}+{c}{\left(−{x}\right)}{x}^{{{2}}}={a}{\left({1}+{x}\right)}+{b}{\left({2}+{x}^{{{2}}}\right)}+{c}{\left(−{x}\right)}.$$

Then comparing the coefficients of power of x we get $$\displaystyle{a}+{2}{b}={0}⋯{\left({1}\right)}$$
$$\displaystyle{a}−{c}={0}⋯{\left({2}\right)}$$
$$\displaystyle{2}{b}={1}⋯{\left({3}\right)}.$$
From (3) we have $$b=\frac{1}{2}$$.

From (1) and (2) we have $$\displaystyle{c}=−{2}{b}=−{1}{c}=−{2}{b}=−{1}$$.

Therefore $$a=c=−1$$.

Thus $$x^{2}$$ can be written as a linear combination of $$\displaystyle{\left({1}+{x}\right)},{\left({2}+{x}^{{{2}}}\right)}{\left({1}+{x}\right)},{\left({2}+{x}^{{{2}}}\right)}$$ and $$−x.$$