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## Homework Statement

Let

S = { p [itex]\in[/itex] [itex]P[/itex][itex]_{2}(ℝ) | p(7) = 0 }

Prove that S is a subspace of P[itex]_{2}(ℝ) (the vector space of all polynomials of degree at most 2)

## The Attempt at a Solution

So essentially I have to prove that S is closed under addition, scalar multiplication and that the zero polynomial is in it.

1) Addition:

Let p1 = (x - 7)[itex]^{2}[/itex] = 0

Let p2 = (x - 7) = 0

Clearly p1(7) = 0 and p2(7) = 0

(p1 + p2)(7) = p1(7) + p2(7) = 0 + 0 = 0

Therefore S is closed under addition.

2) Scalar Multiplication:

λ [itex]\in[/itex] ℝ

(λp1)(7) = λp1(7) = λ(0) = 0

I am not sure whether the proof for scalar multiplication is correct or not.

3) Contains the zero polynomial

I do not know how to do this one.

Any help greatly appreciated.

Thanks