Word Count : 388

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We are to check whether $$ B = \{\,1,\ 2x+1,\ (x-1)^2\,\} $$ is a basis for $\mathbb{P}_2$, the vector space of all real polynomials of degree ≤ 2. --- ### Step 1: Recall Basis Conditions A set of polynomials is a basis of $\mathbb{P}_2$ if: 1. It spans $\mathbb{P}_2$ (can represent any quadratic polynomial $a_0 + a_1 x + a_2 x^2$). 2. It is linearly independent. Since $\dim(\mathbb{P}_2) = 3$, we just need to check that the 3 polynomials are linearly independent; if so, they automatically span $\mathbb{P}_2$. --- ### Step 2: Express the Polynomials in Standard _______ ___ ________ ___ ___ ______.
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