 # 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 Brittney Lord 2021-09-11 Answered

Consider the following polynomials in ${P}_{3}$:
$q\left(x\right)=-{x}^{3}+3{x}^{2}-x+5$
$r\left(x\right)=-4{x}^{3}+7{x}^{2}-x+10$
$u\left(x\right)=-5{x}^{3}+8{x}^{2}+10$
For each of the following polynomials, determine whether it is in {q,r,u}. If so, ecpress it as a lineare combination of the polynomials above. Use e.g. q rather than q(x) to represebt the polynomials in your linear combination.
${p}_{1}\left(x\right)=-3{x}^{2}-3x+10$

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