Linear Algebra

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Difficult

Polynomial Linear Combination

LINALG-XIPSYZ

Suppose that $p(t)=t^3+2t+3$ and $q(t)=6t^3-3t$. Suppose that $r(t)$ is a linear combination of $p(t)$ and $q(t)$.

Which of the following must be true about $r(t)$?

Select ALL that apply.

A

The degree of $r(t)$ is $3$.

B

The degree of $r(t)$ is less than or equal to $9$.

C

The degree of $r(t)$ is less than or equal to $3$.

D

$r(0)$ is a multiple of $3$.

E

If $r(t)$ is not the identically zero polynomial than the degree of $r(t)$ is $3$ or the degree of $r(t)$ is $1$.

F

None of the above must be true.