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[Solution Library] Recall that we use the symbol R[ x ] to mean the real vector space of all polynomials in x with real coefficients. Show that the set p(x)

Question: Recall that we use the symbol $\mathbb{R}\left[ x \right]$ to mean the real vector space of all polynomials in x with real coefficients. Show that the set

\[\left\{ p\left( x \right)\in \mathbb{R}\left[ x \right]:\,\,p\left( 1 \right)=0\text{ and }\deg \left( p\left( x \right) \right)\le 3 \right\}\]

is a subspace of $\mathbb{R}\left[ x \right]$. Is it equal to the span of $x-1$, ${{\left( x-1 \right)}^{2}}$, and ${{\left( x-1 \right)}^{3}}$ ? Explain which steps must be followed in order to answer this question.

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[Solution Library] Recall that we use the symbol R[ x ] to mean the real vector space of all polynomials in x with real coefficients. Show that the set p(x)

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