Consider the following polynomials in P_3:q(x)=-x^3+3x^2-x+5, r(x)=-4x^3+7x^2-x+10, u(x)=-5x^3+8x^2+10

Given the polynomials ${\displaystyle q\left(x\right)=-x^3+3x^2-x+5,r\left(x\right)=-4x^3+7x^2-x+10}$ and ${\displaystyle u\left(x\right)=-5x^3+8x^2}$.
Now, a polynomial p(x) is in span of {q,r,u} if there exist nonzero constants a,b and c such that ${\displaystyle p\left(x\right)=aq+br+cu}$
The polynomial is ${\displaystyle p_1\left(x\right)=-3x^2-3x+10}$
Then,
${\displaystyle p_1=aq+br+cu}$
${\displaystyle -3x^2-3x+10=a(-x^3+3x^2-x+5)+b(-4x^3+7x^2-x+10)+c(-5x^3+8x^2+10)-3x^2-3x+10}$
${\displaystyle =(a-4b-5c)x^3+(3a+7b+8c)x^2+(-a-b)x+5a+10b+10c}$
Equate the coefficients of both sides and find the values of a, b and c.
${\displaystyle a-4b-5c=0}$ (1)
${\displaystyle 3a+7b+8c=-3}$ (2)
${\displaystyle -a-b=-3}$ (3)
${\displaystyle 5a+10b+10c=10}$ (4)
From equation (3),
${\displaystyle a=-3+b}$
${\displaystyle a=3-b}$
Substitute ${\displaystyle b=3-a}$ in equation (1) and equation (2) and (3)
${\displaystyle -(3-b)-4b-5c=0}$ (4)
${\displaystyle 3(3-b)+7b+8c=-3}$ (5)
${\displaystyle 5(3-b)+10b+10c=10}$ (6)
From equation (4),
${\displaystyle -3+b-4b-5c=0}$
${\displaystyle -3-3b-5c=0}$
${\displaystyle -3b=5c+3}$
${\displaystyle b=-\frac{5c+3}{3}}$
Substitute ${\displaystyle b=-\frac{5c+3}{3}}$ and in equation (5) and (6)
${\displaystyle -9+4(-\frac{5c+3}{3})+8c=-3}$ (7)
${\displaystyle 15+5(-\frac{5c+3}{3})+10c=10}$

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