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

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

${x}^{3}-2{x}^{2}-5x-3,$

$3{x}^{3}-5{x}^{2}-4x-9$

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The given polynomials are
$2{x}^{3}-2{x}^{2}+12x-6$
${x}^{3}-2{x}^{2}-5x-3$
$3{x}^{3}-5{x}^{2}-4x-9$
The polynomial can be written as vector form
${v}_{1}=\left[\begin{array}{c}2\\ -2\\ 12\\ -6\end{array}\right],{v}_{2}=\left[\begin{array}{c}1\\ -2\\ -5\\ -3\end{array}\right],{v}_{2}=\left[\begin{array}{c}3\\ -5\\ -4\\ -9\end{array}\right]$
Let, ${v}_{1}=a{v}_{2}+b{v}_{3}$
$\left[\begin{array}{c}2\\ -2\\ 12\\ -6\end{array}\right]=a\left[\begin{array}{c}1\\ -2\\ -5\\ -3\end{array}\right]+b\left[\begin{array}{c}3\\ -5\\ -4\\ -9\end{array}\right]$
$\left[\begin{array}{c}2\\ -2\\ 12\\ -6\end{array}\right]=\left[\begin{array}{c}a+3b\\ -2a-5b\\ 5a-4b\\ -3a-9b\end{array}\right]$
$⇒a+3b=2⇒-5a-4b=12$
$⇒-2a-5b=-2⇒-3a-9b=-6$
using equation
$6b-5b=4-2$
$b=2$
$a=-4$
These values of a,b are also santisty, hence polynomial having linear relationship
${P}_{1}=-4{P}_{2}+2{P}_{3}$
or $2{P}_{3}={P}_{1}+4{P}_{2}$
${P}_{1}=$ I polynomial
${P}_{2}=$ II polynomial
${P}_{3}=$ III polynomial