# Determine whether the first polynomial can be expressed as a linear combination

Determine whether the first polynomial can be expressed as a linear combination of other two polynomials.
$$\displaystyle{2}{x}^{{3}}-{2}{x}^{{2}}+{12}{x}-{6},{x}^{{3}}-{2}{x}^{{2}}-{5}{x}-{3},{3}{x}^{{3}}-{5}{x}^{{2}}-{4}{x}-{9}$$

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odgovoreh

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