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# Derive some properties of the product of ex with a polynomial

Let 1. Prove that where denotes the th derivative of .

2. Do part (a) in the case that is a cubic polynomial.
3. Find a similar formula and prove it in the case that is a polynomial of degree .

For all of these we recall from a previous exercise (Section 5.11, Exercise #4) that by Leibniz’s formula if then the th derivative is given by So, in the case at hand we have and so (Since the th derivative of is still for all and .)

1. Proof. From the formula above we have But, since is a quadratic polynomial we have Hence, we have 2. If is a cubic polynomial we may write, Claim: If then Proof. We follow the exact same procedure as part (a) except now we have the derivatives of given by Therefore, we now have 3. Claim: Let be a polynomial of degree , Let . Then, Proof. Using Leibniz’s formula again, we have But for the degree polynomial , we know if and for all . Hence, we have 