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I want to know how to solve the question if you have a set of polynomials

$$ p_1 = 1 + x, $$ $$ p_2 = 1 + 2x + x^2, $$ $$ p_3 = 1 + 3x + 3x^2 + x^3, $$ and if the polynomial $p = 2017 + x^2 + 26x^3$ is spanned by the set $S = \{p_1,p_2,p_3\}$

Thank you

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    What does it mean for a polynomial to span a set? If you are somehow talking about vector spaces....well, $S$ looks three dimensional so how could one vector span it? – 2017-02-17
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    Is it possible that you mean to ask: is $p(x)$ in the linear span of $\{p_1,p_2,p_3\}$ (maybe over $\mathbb R$....who knows?).? If so, then you have a system of linear equations. Write $p(x)=Ap_1+Bp_2+Cp_3$ and note that $A+B+C=2017$ and so on. Helps to remark that $C=26$. – 2017-02-17
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    You should add more information like which is the field (if is a vector space) or the ring (if is a module)? – 2017-02-17
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    Thank you for quick response. Yes it is a vector space – 2017-02-17

2 Answers 2

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If the polynomials are over $\mathbb{R}$, the question is solved by a linear system given by:

As $1,x,x^2,x^3$ are l.i. then the eaquation $p=ap_1+bp_2+cp_3$ is solved by$$\begin{cases} a+b+c=2017\\a+2b+3c=0\\b+3c=1\\c=26\end{cases}$$

So as the system don't have a solution the answer is no.

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Substitute $x+1=t$

Then $p=26(t-1)^3 +(t-1)^2 +2017=26t^3-77t^2+76t+1992$

Since there exists a constant, $p$ cannot be expressed as a linear combination of $t, t^2, t^3$