Given \(\displaystyle{P}{\left({x}\right)}={3}{x}^{{2}}+{4}{y}^{{2}}\) and \(\displaystyle{R}{\left({x}\right)}=-{7}{x}^{{2}}+{4}{x}{y}-{3}{y}^{{2}}\), find \(P(x)-R(x)\).

\((2x − 1)\) is a factor of the polynomial \(\displaystyle{6}{x}^{{6}}+{x}^{{5}}-{92}{x}^{{4}}+{45}{x}^{{3}}+{184}{x}^{{2}}+{4}{x}-{48}\). Determine whether the statement is true or false. Justify your answer.

Do the polynomials \(\displaystyle{x}^{{3}}-{2}{x}^{{2}}+{1},{4}{x}^{{2}}-{x}+{3}\), and \(3x-2\) generate \(\displaystyle{P}_{{3}}{\left({R}\right)}\)? Justify your answer.

Determine whether the following polynomials u,v, w in P(t) are linearly dependent or independent: \(\displaystyle{u}={t}^{{3}}-{4}{t}^{{2}}+{3}{t}+{3},\)

\({v}={t}^{{3}}+{2}{t}^{{2}}+{4}{t}-{1},\)

\({w}={2}{t}^{{3}}-{t}^{{2}}-{3}{t}+{5}\)

Find the product of \(\displaystyle{2}{x}^{{3}}+{3}{x}^{{2}}+{1}\) and \(\displaystyle{2}{x}^{{2}}+{4}\) in \(z_5[x]/\)

a) Find the derivative of \(f(x)=(2x+9)(8x−7)\) by first expanding the polynomials. b) Find the derivative of \(f(x)=(2x+9)(8x−7)\) by using the product rule. Let \(\displaystyle g{{\left({x}\right)}}={2}{x}+{9}{\quad\text{and}\quad}{h}{\left({x}\right)}={8}{x}-{7}\)