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

Question

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

FobelloE 2021-09-03 Answered

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

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

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

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Expert Answer

timbalemX

Answered 2021-09-04 Author has 108 answers

The given polynomials are
$2 x^{3} - 2 x^{2} + 12 x - 6$
$x^{3} - 2 x^{2} - 5 x - 3$
$3 x^{3} - 5 x^{2} - 4 x - 9$
The polynomial can be written as vector form
$v_{1} = [\begin{matrix} 2 \\ - 2 \\ 12 \\ - 6 \end{matrix}], v_{2} = [\begin{matrix} 1 \\ - 2 \\ - 5 \\ - 3 \end{matrix}], v_{2} = [\begin{matrix} 3 \\ - 5 \\ - 4 \\ - 9 \end{matrix}]$
Let, $v_{1} = a v_{2} + b v_{3}$
$[\begin{matrix} 2 \\ - 2 \\ 12 \\ - 6 \end{matrix}] = a [\begin{matrix} 1 \\ - 2 \\ - 5 \\ - 3 \end{matrix}] + b [\begin{matrix} 3 \\ - 5 \\ - 4 \\ - 9 \end{matrix}]$
$[\begin{matrix} 2 \\ - 2 \\ 12 \\ - 6 \end{matrix}] = [\begin{matrix} a + 3 b \\ - 2 a - 5 b \\ 5 a - 4 b \\ - 3 a - 9 b \end{matrix}]$
$\Rightarrow a + 3 b = 2 \Rightarrow - 5 a - 4 b = 12$
$\Rightarrow - 2 a - 5 b = - 2 \Rightarrow - 3 a - 9 b = - 6$
using equation
$6 b - 5 b = 4 - 2$
$b = 2$
$a = - 4$
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