Question

Determine whether the first polynomial can be expressed as a linear combination of other two polynomials.
\(\displaystyle{2}{x}^{{3}}-{2}{x}^{{2}}+{12}{x}-{6},\)

\({x}^{{3}}-{2}{x}^{{2}}-{5}{x}-{3},\)

\({3}{x}^{{3}}-{5}{x}^{{2}}-{4}{x}-{9}\)

Answer 1

The given polynomials are
\(\displaystyle{2}{x}^{{3}}−{2}{x}^{{2}}+{12}{x}−{6}\)
\(\displaystyle{x}^{{3}}−{2}{x}^{{2}}−{5}{x}−{3}\)
\(\displaystyle{3}{x}^{{3}}−{5}{x}^{{2}}−{4}{x}−{9}\)
The polynomial can be written as vector form
\(v_1=\begin{bmatrix}2\\-2\\12\\-6\end{bmatrix},v_2=\begin{bmatrix}1\\-2\\-5\\-3\end{bmatrix},v_2=\begin{bmatrix}3\\-5\\-4\\-9\end{bmatrix}\)
Let, \(\displaystyle{v}_{{1}}={a}{v}_{{2}}+{b}{v}_{{3}}\)
\(\begin{bmatrix}2\\-2\\12\\-6\end{bmatrix}=a\begin{bmatrix}1\\-2\\-5\\-3\end{bmatrix}+b\begin{bmatrix}3\\-5\\-4\\-9\end{bmatrix}\)
\(\begin{bmatrix}2\\-2\\12\\-6\end{bmatrix}=\begin{bmatrix}a+3b\\-2a-5b\\5a-4b\\-3a-9b\end{bmatrix}\)
\(\displaystyle\Rightarrow{a}+{3}{b}={2}\Rightarrow-{5}{a}-{4}{b}={12}\)
\(\displaystyle\Rightarrow-{2}{a}-{5}{b}=-{2}\Rightarrow-{3}{a}-{9}{b}=-{6}\)
using equation
\(\displaystyle{6}{b}-{5}{b}={4}-{2}\)
\(b=2\)
\(a=-4\)
These values of a,b are also santisty, hence polynomial having linear relationship
\(\displaystyle{P}_{{1}}=-{4}{P}_{{2}}+{2}{P}_{{3}}\)
or \(\displaystyle{2}{P}_{{3}}={P}_{{1}}+{4}{P}_{{2}}\)
\(\displaystyle{P}_{{1}}=\) I polynomial
\(\displaystyle{P}_{{2}}=\) II polynomial
\(\displaystyle{P}_{{3}}=\) III polynomial