Exercise · ZW-POLYNOMIAL-RING-010 · ★★★ Advanced
How many polynomials of degree exactly 2 are there in \(\mathbb{Z}_{3}[x]\)?
Choose one option.
Show answer
- C. \(18\)
Explanation
The coefficient of \(x^2\) has 2 choices (nonzero) and those of \(x\) and the constant term have 3 each: \(2\cdot3\cdot3=18\). The 27 would include those of lower degree.
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-008 In ℤ_2[x], what is (x+1)^4?