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## Homework Statement

Find a basis of U, the subspace of P

_{3}

U = {p(x) in P

_{3}| p(7) = 0, p(5) = 0}

## Homework Equations

## 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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