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

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

Question
Polynomials
asked 2021-01-13
write the polynomial \(\displaystyle{P}{\left({x}\right)}={x}^{{{2}}},\) if possible as a linear combination of the polynomials \(\displaystyle{1}+{x},{2}+{x}^{{{2}}},−{x}.\)

Answers (1)

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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