1. The problem asks to determine if $\langle P_{3}(\mathbb{R}), +, \cdot \rangle$ is a vector space. 2. $P_{3}(\mathbb{R})$ is the set of all real polynomials of degree at most 3. 3. A vector space requires that the set with addition and scalar multiplication satisfy 10 axioms (closure, associativity, identity, inverse for addition, distributivity, scalar identity, etc.). 4. Polynomials of degree at most 3 are closed under addition because sum of two polynomials degree at most 3 also has degree at most 3. 5. They are closed under scalar multiplication because multiplying by a real scalar does not increase the degree. 6. The zero polynomial (all zero coefficients) serves as the additive identity. 7. Every polynomial has an additive inverse (negating all coefficients). 8. Addition and scalar multiplication are associative and distributive by the properties of polynomial addition and multiplication. 9. The scalar identity holds because multiplying by 1 leaves polynomials unchanged. 10. Therefore, all vector space axioms hold, proving $\langle P_{3}(\mathbb{R}), +, \cdot \rangle$ is a vector space. Final answer: Yes, $\langle P_{3}(\mathbb{R}), +, \cdot \rangle$ is a vector space.