(b) Generalize the remainder theorem to work for lm.d if m 7 0 instead of only l1,-k. In other words, what is the remainder after dividing f by lm,d if m # 0?

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Theorem 4.15. The remainder theorem
If f is any polynomial and k is a real number, then there is a unique polynomial q such that
f = q •l1,-k + const f(k)·
Proof. This follows directly from division of polynomials. Because the degree of l1,-
Theorem 4.13, there is a unique polynomial q and a unique constant r such that
is
one, from
f = q •l1,-k + r.
Then
f (k) = q(k)(k – k) + r = r.
|
57
Exercise 4.16.
(a) Find q in the conclusion of the remainder theorem.
(b) Generalize the remainder theorem to work for lm.d if m + 0 instead of only l1,-k. In other
words, what is the remainder after dividing f by lm,d if m # 0?
Transcribed Image Text:Theorem 4.15. The remainder theorem If f is any polynomial and k is a real number, then there is a unique polynomial q such that f = q •l1,-k + const f(k)· Proof. This follows directly from division of polynomials. Because the degree of l1,- Theorem 4.13, there is a unique polynomial q and a unique constant r such that is one, from f = q •l1,-k + r. Then f (k) = q(k)(k – k) + r = r. | 57 Exercise 4.16. (a) Find q in the conclusion of the remainder theorem. (b) Generalize the remainder theorem to work for lm.d if m + 0 instead of only l1,-k. In other words, what is the remainder after dividing f by lm,d if m # 0?
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