\((x+h)^{3}=(x+h)(x+h)^{3}=(x+h)(x^{2}+2hx+h^{2})=x^{3}+2hx^{2}+h^{2}x+hx^{2} +2h^{2}x+h^{3}=x^{3}+3hx^{2}+3h^{2}x+h^{3}\)

asked 2021-09-09

Solve the equation:

\(\displaystyle{\left({3}{x}+{1}\right)}{\left({x}^{{2}}+{2}{x}-{10}\right)}\)

\(\displaystyle{\left({3}{x}+{1}\right)}{\left({x}^{{2}}+{2}{x}-{10}\right)}\)

asked 2021-09-14

Consider the following polynomials over \(\displaystyle{Z}_{{8}}\) where a is written for [a] in \(\displaystyle{Z}_{{8}}\):

\(\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{3}}+{7}{x}+{4},{g{{\left({x}\right)}}}={4}{x}^{{2}}+{4}{x}+{6},{h}{\left({x}\right)}={6}{x}^{{2}}+{3}\)

Find each of the following polynomials with all coefficients in \(\displaystyle{Z}_{{8}}\)

\(\displaystyle{g{{\left({x}\right)}}}+{h}{\left({x}\right)}\)

\(\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{3}}+{7}{x}+{4},{g{{\left({x}\right)}}}={4}{x}^{{2}}+{4}{x}+{6},{h}{\left({x}\right)}={6}{x}^{{2}}+{3}\)

Find each of the following polynomials with all coefficients in \(\displaystyle{Z}_{{8}}\)

\(\displaystyle{g{{\left({x}\right)}}}+{h}{\left({x}\right)}\)

asked 2021-09-08

Solve the equation:

\(\displaystyle{\left({2}{x}-{5}\right)}+{\left({3}{x}^{{2}}+{7}{x}+{10}\right)}\)

\(\displaystyle{\left({2}{x}-{5}\right)}+{\left({3}{x}^{{2}}+{7}{x}+{10}\right)}\)

asked 2021-11-08

Consider the following polynomials in \(\displaystyle{P}_{{3}}\):

\(\displaystyle{q}{\left({x}\right)}=-{x}^{{3}}+{3}{x}^{{2}}-{x}+{5}\)

\(\displaystyle{r}{\left({x}\right)}=-{4}{x}^{{3}}+{7}{x}^{{2}}-{x}+{10}\)

\(\displaystyle{u}{\left({x}\right)}=-{5}{x}^{{3}}+{8}{x}^{{2}}+{10}\)

For each of the following polynomials, determine whether it is in span {q,r,u}. If so, ecpress it as a lineare combination of the polynomials above. Use e.g. q rather than q(x) to represebt the polynomials in your linear combination.

\(\displaystyle{p}_{{1}}{\left({x}\right)}=-{3}{x}^{{2}}-{3}{x}+{10}\)

\(\displaystyle{q}{\left({x}\right)}=-{x}^{{3}}+{3}{x}^{{2}}-{x}+{5}\)

\(\displaystyle{r}{\left({x}\right)}=-{4}{x}^{{3}}+{7}{x}^{{2}}-{x}+{10}\)

\(\displaystyle{u}{\left({x}\right)}=-{5}{x}^{{3}}+{8}{x}^{{2}}+{10}\)

For each of the following polynomials, determine whether it is in span {q,r,u}. If so, ecpress it as a lineare combination of the polynomials above. Use e.g. q rather than q(x) to represebt the polynomials in your linear combination.

\(\displaystyle{p}_{{1}}{\left({x}\right)}=-{3}{x}^{{2}}-{3}{x}+{10}\)

asked 2021-09-16

asked 2021-09-12

Subtract: \(\displaystyle{\left({14}{x}^{{3}}-{5}{x}^{{2}}+{x}-{9}\right)}-{\left({4}{x}^{{3}}-{3}{x}^{{2}}-{7}{x}+{1}\right)}\)

asked 2021-09-13

Add polynomials:

\(\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{3}}+{5}{x}^{{2}}-{3}{x}+{4}\)

\(\displaystyle{g{{\left({x}\right)}}}={x}^{{5}}-{2}{x}^{{2}}-{1}\)

\(\displaystyle{f{{\left({x}\right)}}}+{g{{\left({x}\right)}}}=\)

\(\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{3}}+{5}{x}^{{2}}-{3}{x}+{4}\)

\(\displaystyle{g{{\left({x}\right)}}}={x}^{{5}}-{2}{x}^{{2}}-{1}\)

\(\displaystyle{f{{\left({x}\right)}}}+{g{{\left({x}\right)}}}=\)