Exercise · ZW-POLYNOMIAL-DIVISION-ALGORITHM-005 · ★☆☆ Basic
In \(\mathbb{Z}_5[x]\), what are the quotient \(q\) and remainder \(r\) of dividing \(x^{3} + 2 x + 1\) by \(2 x + 3\)?
Choose one option.
Show answer
- B. \(q=3 x^{2} + 3 x + 4,\ r=4\)
Explanation
One checks that \((2x+3)(3x^2+3x+4)+4\) equals \(x^3+2x+1\) modulo 5. The other pairs fail the equality. Since \(\mathbb{Z}_5\) is a field, the quotient and remainder are unique.
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-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-008 Compute the remainder of dividing x^5 + 3 x^3 - x + 2 by x^2 + 1 in ℚ[x].
- 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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