Exercise · ZW-POLYNOMIAL-RING-008 · ★★☆ Intermediate
In \(\mathbb{Z}_{2}[x]\), what is \((x+1)^4\)?
Choose one option.
Show answer
- D. \(x^4+1\)
Explanation
First \((x+1)^2=x^2+2x+1=x^2+1\) in \(\mathbb{Z}_{2}[x]\), and squaring again \((x^2+1)^2=x^4+1\).
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
- ZW-POLYNOMIAL-RING-001 In ℤ[x], what is (x+1)(x+2)?
- ZW-POLYNOMIAL-RING-002 In ℤ_3[x], what is (x+1)(x+2)?
- 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-007 In ℤ_6[x], what is the degree of (2x+1)(3x+1)?
- ZW-POLYNOMIAL-RING-009 Which of these polynomials is a unit of ℤ_4[x]?