How do you reduce the coefficients of the polynomial $f(x) = x^4 - x^3 + x^2 - 2x + 4$ modulo 3 to find $\bar{f}(x)$ ?

Q: How do you reduce the coefficients of the polynomial f(x) = x^4 - x^3 + x^2 - 2x + 4 modulo 3 to find \bar{f}(x)?

To find $\bar{f}(x)$, you reduce each coefficient of $f(x)$ modulo 3. The coefficient of $x^4$ is 1, which remains 1. The coefficient of $x^3$ is -1, which is congruent to 2 mod 3. The coefficient of $x^2$ is 1, which remains 1. The coefficient of $x$ is -2, which is congruent to 1 mod 3. The constant term is 4, which is congruent to 1 mod 3. Thus, $\bar{f}(x) = x^4 + 2x^3 + x^2 + x + 1$ in $\mathbb{Z}_3[x]$.

To find

\bar{f}(x)

, you reduce each coefficient of

f(x)

modulo 3. The coefficient of

x^4

is 1, which remains 1. The coefficient of

x^3

is -1, which is congruent to 2 mod 3. The coefficient of

x^2

is 1, which remains 1. The coefficient of

x

is -2, which is congruent to 1 mod 3. The constant term is 4, which is congruent to 1 mod 3. Thus,

\bar{f}(x) = x^4 + 2x^3 + x^2 + x + 1

\mathbb{Z}_3[x]

Conditions

The polynomial is $f(x) = x^4 - x^3 + x^2 - 2x + 4$ .
Coefficients are reduced modulo 3.

Reasoning, step by step

Identify the coefficient of $x^4$ as 1, which is $1 \pmod 3$ .
Identify the coefficient of $x^3$ as -1, which is $2 \pmod 3$ .
Identify the coefficient of $x^2$ as 1, which is $1 \pmod 3$ .
Identify the coefficient of $x$ as -2, which is $1 \pmod 3$ .
Identify the constant term as 4, which is $1 \pmod 3$ .
Construct the new polynomial $\bar{f}(x) = x^4 + 2x^3 + x^2 + x + 1$ .

Example

The narrator states, 'The coefficient for x to the fourth is 1... For x cubed, it's negative 1, which is congruent to 2 mod 3... For x squared, it's 1... For x, it's negative 2, which is congruent to 1 mod 3. And the constant term is 4, which is 1 mod 3.'

Common misconceptions

Assuming negative coefficients remain negative in $\mathbb{Z}_3$ instead of converting them to positive equivalents (e.g., -1 to 2).
Forgetting to reduce the constant term modulo 3.

Watch the explanation

YouTubeMastering Prime Ideals in Quotient Rings Step by Step Abstract Algebra Tutorial

1:32 – 2:00Watch this moment ↗

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How do you reduce the coefficients of the polynomial f(x)=x4−x3+x2−2x+4f(x) = x^4 - x^3 + x^2 - 2x + 4f(x)=x4−x3+x2−2x+4 modulo 3 to find fˉ(x)\bar{f}(x)fˉ​(x)?