Polynomial long division
Dividing a polynomial A by a non-zero polynomial B: A = BQ + R with a remainder of degree strictly less than that of B. The remainder of division by (x − a), divisibility and Horner's scheme.
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Polynomial division with remainder: what is it?
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For A and B ≠ 0, there is a unique pair (Q, R) such that A = BQ + R and R is zero or of degree strictly less than that of B.
As for integers (17 = 5 × 3 + 2 with 2 < 5), the condition on the remainder makes the pair unique. Q is found term by term, dividing the highest-degree terms at each step.
What is the remainder of dividing P(x) = x³ − 2x + 1 by x + 1?
- 2
- 0
- −2
- 4
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2
x + 1 = x − (−1), so the remainder is P(−1) = −1 + 2 + 1 = 2. The decoy 0 comes from P(1) = 0, taking the wrong root; −2 is a sign slip.
In A = BQ + R, the degree of R is … than that of B, and deg Q = … when deg A ≥ deg B.
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In A = BQ + R, the degree of R is strictly less than that of B, and deg Q = deg A − deg B when deg A ≥ deg B.
Dividing by a polynomial of degree 2 leaves a remainder of degree at most 1; a polynomial of degree 5 divided by one of degree 2 gives a quotient of degree 3.
Remainder theorem: what is it?
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The remainder of dividing a polynomial P by (x − a) is the number P(a).
For P(x) = x³ + 2x + 1 divided by x − 2, the remainder is P(2) = 8 + 4 + 1 = 13, without doing the division. And P(a) = 0 is equivalent to: (x − a) divides P.