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

Question
Polynomials
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=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.

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