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 P3:
q(x)=x3+3x2x+5
r(x)=4x3+7x2x+10
u(x)=5x3+8x2+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.
p1(x)=3x23x+10

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Expert Answer

okomgcae
Answered 2021-09-12 Author has 93 answers

Given the polynomials q(x)=x3+3x2x+5,r(x)=4x3+7x2x+10 and u(x)=5x3+8x2.
Now, a polynomial p(x) is in span of {q,r,u} if there exist nonzero constants a,b and c such that p(x)=aq+br+cu
The polynomial is p1(x)=3x23x+10
Then,
p1=aq+br+cu
3x23x+10=a(x3+3x2x+5)+b(4x3+7x2x+10)+c(5x3+8x2+10)3x23x+10
=(a4b5c)x3+(3a+7b+8c)x2+(ab)x+5a+10b+10c
Equate the coefficients of both sides and find the values of a, b and c.
a4b5c=0 (1)
3a+7b+8c=3 (2)
ab=3 (3)
5a+10b+10c=10 (4)
From equation (3),
a=3+b
a=3b
Substitute b=3a in equation (1) and equation (2) and (3)
(3b)4b5c=0 (4)
3(3b)+7b+8c=3 (5)
5(3b)+10b+10c=10 (6)
From equation (4),
3+b4b5c=0
33b5c=0
3b=5c+3
b=5c+33
Substitute b=5c+33 and in equation (5) and (6)
9+4(5c+33)+8c=3 (7)
15+5(5c+33)+10c=10

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