Number of polynomials

Task number: 2633

Determine how many polynomials \(p\) in \(\mathbb Z_5\) of degree three satisfy \(p(3)=2\). Describe all such polynomials.

  • Resolution

    For a possible \(p(x)=a_3x^3+a_2x^2+a_1x+a_0\) we have four choices for \(a_3\), namely: \(1,…,4\); for \(a_2\) and \(a_1\) five choices: \(0,…,4\); and the value of \(a_0\) can be determined when \(x\) is subtituted by 3. In total there are hundred of such polynomials.

    The set of the desired polynomials can be described as:
    \(\{a_3x^3+a_2x^2+a_1x+(3a_3+a_2+2a_1+2): a_3\in\{1,…,4\}, a_2, a_1\in\{0,…,4\}\}\).

Difficulty level: Easy task (using definitions and simple reasoning)
Proving or derivation task
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