Exercise · ZW-POLYNOMIAL-RING-007 · ★★☆ Intermediate
In \(\mathbb{Z}_{6}[x]\), what is the degree of \((2x+1)(3x+1)\)?
Choose one option.
Show answer
- A. \(1\)
Explanation
The product is \(6x^2+5x+1\), and since \(6=0\) in \(\mathbb{Z}_{6}\) it becomes \(5x+1\), of degree 1. Degrees do not add because \(\mathbb{Z}_{6}\) has zero divisors.
Review the concept: Polynomial ring
The polynomial ring \(R[x]\) over a commutative ring \(R\) with identity consists of the finite sums of terms \(a_ix^i\), with coefficients \(a_i\in R\) and an indeterminate \(x\).
\[\Big(\sum_i a_ix^i\Big)\Big(\sum_j b_jx^j\Big)=\sum_k\Big(\sum_{i+j=k}a_ib_j\Big)x^k\]
More practice problems: polynomial ring
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- ZW-POLYNOMIAL-RING-003 In ℤ[x], what is the coefficient of x^2 in (1+x+x^2)(2+x+3x^2)?
- ZW-POLYNOMIAL-RING-004 In ℤ_2[x], what is (x^2+x+1)+(x+1)?
- ZW-POLYNOMIAL-RING-005 Which statement about ℤ[x] is true?
- ZW-POLYNOMIAL-RING-006 In ℤ[x], f has degree 3 and g has degree 4. What is the degree of fg?
- ZW-POLYNOMIAL-RING-008 In ℤ_2[x], what is (x+1)^4?
- ZW-POLYNOMIAL-RING-009 Which of these polynomials is a unit of ℤ_4[x]?