(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find a basis of U, the subspace of P_{3}

U = {p(x) in P_{3}| p(7) = 0, p(5) = 0}

2. Relevant equations

3. The attempt at a solution

ax^{3}+bx^{2}+cx+d

p(7)=343a+49b+7c+d=0

p(5)=125a+25b+5c+d=0

d=-343a-49b-7c

d=-125a-25b-5c

ax^{3}+bx^{2}+cx+{(d+d)/2} -->{(d+d)/2}=2d/2=d

(-343a-49b-7c-125a-25b-5c)/2=-234a-37b-6c

ax^{3}+b^{2}+cx-234a-37b-6c

a(x^{3}-234)+b(x^{2}-37)+c(x-6)

basis{x^{3}-234,x^{2}-37,x-6}

dim=3

please check if I m correct or not

and is there a easier way to do it?

also, if

U = {p(x) in P_{3}| p(7) = 0, p(5) = 0,p(3) = 0,p(1) = 0}

p(x) does not exist?

thanks!

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# Find a basis of U, the subspace of P3

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