Exercise · ZW-POLYNOMIAL-DIVISION-ALGORITHM-008 · ★★☆ Intermediate
Compute the remainder of dividing \(x^{5} + 3 x^{3} - x + 2\) by \(x^{2} + 1\) in \(\mathbb{Q}[x]\).
Choose one option.
Show answer
- A. \(2 - 3 x\)
Explanation
Since \(x^2\equiv-1\), we get \(x^5\equiv x\) and \(x^3\equiv-x\), so \(f\equiv x-3x-x+2=-3x+2\). It has degree less than 2, as the algorithm requires.
Review the concept: Polynomial division algorithm
Given polynomials \(f\) and \(g\neq 0\) over a field \(F\), there exist a unique quotient \(q\) and a unique remainder \(r\) satisfying the equality below, with \(r\) zero or of degree less than that of \(g\).
\[f=gq+r,\qquad r=0\ \text{ or }\ \deg r<\deg g\]
More practice problems: polynomial division algorithm
- ZW-POLYNOMIAL-DIVISION-ALGORITHM-001 When dividing x^3 - 2 x^2 + x - 5 by x-2 in ℚ[x], what is the remainder?
- ZW-POLYNOMIAL-DIVISION-ALGORITHM-002 Compute the quotient of dividing 2 x^3 + 3 x^2 - x + 4 by x+1 in ℚ[x].
- ZW-POLYNOMIAL-DIVISION-ALGORITHM-003 When dividing f by a polynomial g of degree 3 over a field, what is known about the remainder r?
- ZW-POLYNOMIAL-DIVISION-ALGORITHM-004 Over ℚ, what is the division of x^4 + 1 by x^2 + x + 1?
- ZW-POLYNOMIAL-DIVISION-ALGORITHM-005 In ℤ_5[x], what are the quotient q and remainder r of dividing x^3 + 2 x + 1 by 2 x + 3?
- ZW-POLYNOMIAL-DIVISION-ALGORITHM-006 What is the remainder of dividing x^101+x^2+1 by x+1 in ℚ[x]?
- ZW-POLYNOMIAL-DIVISION-ALGORITHM-007 In ℤ[x], we want to divide x^2 by 2x with r=0 or deg r<1, and q,r∈ℤ[x]. What happens?
- ZW-POLYNOMIAL-DIVISION-ALGORITHM-009 In the uniqueness proof, if f=gq_1+r_1=gq_2+r_2 with q_1ne q_2, what contradiction appears?
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