Which of the following sets is a subspace of set of prime numbers P Subscript nℙn for an appropriate value of n? A. All polynomials of the form p (t) equals=aplus+bt squaredt2 , where a and b are in set of real numbers Rℝ B. All polynomials of degree exactly 4, with real coefficients C. All polynomials of degree at most 4, with positive coefficients

Question

Mathematics

Trevon

9 December, 19:33

Which of the following sets is a subspace of set of prime numbers P Subscript nℙn for an appropriate value of n? A. All polynomials of the form p (t) equals=aplus+bt squaredt2 , where a and b are in set of real numbers Rℝ B. All polynomials of degree exactly 4, with real coefficients C. All polynomials of degree at most 4, with positive coefficients

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Dayami Navarro · Answer 1

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Step-by-step explanation:

A. All polynomials of the form p (t) = a + bt2, where a and b are in: This means that A is closed under scalar mult and vector addition, and includes the zero vector.

B. All polynomials of degree exactly 4, with real coefficients: what this means is that under vector addition, B isn't closed, and it does not consist of the zero vector. What it consist of is just polynomials with degree exactly 4. Let f=x4+1f=x4+1 and let g=-x4g=-x4. Both are in B, but their sum is not, because it has degree 0.

C. All polynomials of degree at most 4, with positive coefficients: what this means is that C is not a subspace for the reason that the positive coefficients make zero vector impossible. The restriction there also makes C not closed under multiplication by the scalar - 1.

So the answer is only A : D