Determining if a Set of Polynomials is a Subspace of ℙn by Checking Three Criteria

Determine if the given set is a subspace of ℙn. Justify your answer.The set of all polynomials in ℙn such that p​(0)=0

To determine if a set is a subspace of ℙn, we need to check if it satisfies three criteria:

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To determine if a set is a subspace of ℙn, we need to check if it satisfies three criteria:

1. The set is non-empty.
2. The set is closed under addition.
3. The set is closed under scalar multiplication.

Let’s analyze each criterion:

1. Non-empty:
We can say that the set is non-empty because the zero polynomial, denoted as 0, belongs to the set of all polynomials in ℙn, and we have p​(0) = 0 for the zero polynomial.

2. Closure under addition:
To check if the set is closed under addition, we need to take any two polynomials, p(x) and q(x), from the set and see if their sum also belongs to the set.

Let p(x) and q(x) be two polynomials such that p​(0) = q(0) = 0.
Then, for the sum r(x) = p(x) + q(x), we have r​(0) = p(0) + q(0) = 0 + 0 = 0.

Since r(0) is also equal to 0, we can conclude that the set is closed under addition.

3. Closure under scalar multiplication:
To check if the set is closed under scalar multiplication, we need to take any polynomial p(x) from the set and see if the product of p(x) and a scalar c also belongs to the set.

Let p(x) be a polynomial such that p(0) = 0 and c be a scalar.
Then, for the product q(x) = c * p(x), we have q(0) = c * p(0) = c * 0 = 0.

Since q(0) is also equal to 0, we can conclude that the set is closed under scalar multiplication.

Based on our analysis, we can say that the given set of all polynomials in ℙn such that p​(0) = 0 is indeed a subspace of ℙn.

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