(2) There is a Division Algorithm for Polynomials that states the following: Giv
ID: 3115553 • Letter: #
Question
(2) There is a Division Algorithm for Polynomials that states the following: Given any two non-constant polynomials p1) and p2x) where the degree of p1(x) is greater than that of p2(x), there exist polynomials q(x) and r(x) such that p1(x) = p2(x)q(x) + r(x), where either r(x) is the zero polynomial, or the degree of r(x) is strictly less than the degree of p2(x). Put another way, this means that p1(x)/P2(x) = q(x) + r(x)/P2(x).] Use this result to prove the following key step in the proof of the Fundamental Theorem of Algebra: If a is a zero of the polynomial px), then (x - a) is a factor of px).Explanation / Answer
Ans:
Step 1: Given polynomial p(x). We need to prove if a is a zero of p(x) then (x - a) is a factor of p(x)
Step 2: Since a is a zero of p(x), then p(a) = 0
Step 3: Now, we can write p(x) as
p(x) = (x - a)*q(x) + r(x)
Degree of r(x) should be less than degree of (x - a)
Therefore, degree of r(x) should be 0
Hence r(x) is a constant. Let's right it as r
p(x) = (x - a)q(x) + r
Since p(a) = 0
0 = (a - a)q(x) + r
Therefore, r = 0
Step 4: Therefore we can write
p(x) = (x - a)q(x)
Hence (x - a) is a factor of p(x) when a is a zero of p(x)
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.