Polynomial Division and Modular Arithmetic: Remainder When Divided by x + 2

Math Problem Statement

Turn #1 - Prompt Read-only

Consider this math question: Let ( p(x) ) be a polynomial with integer coefficients, and let ( a ) be an integer such that ( p(a) \equiv 0 \pmod{5} ). If ( p(x) ) is divided by ( (x - a)(x + 2) ), the remainder is ( 3x + 4 ). Find the remainder when ( p(x) ) is divided by ( x + 2 ). 53 words Answer Provide the final answer in the simplest form.

Can the prompt be solved? *

Select Yes if the given prompt can be solved or has a clear solution.

Choose No if the prompt cannot be solved or lacks sufficient information to provide a solution. This helps to indicate whether the task is feasible.

REMEMBER: if the prompt is not solvable, then the hint should direct the model to the realization that it’s unsolvable.

YES

NO What is the final answer to the given problem? *

Determine the correct mathematical answer for the prompt above. The final answer should be in the simplest form (e.g., only include the answer; complete sentences are unnecessary; this should rarely be longer than a single line).

(\boxed{-2}) 2 words

LaTex Preview:

− 2 −2

Save and Continue Hint Write the hint for the provided problem. Generally, this includes clarifications of the setup, summaries of a planned approach, and implementation steps. For Math, the hint should be similar to reasoning while addressing specific techniques and methods relevant to the problem. For Code, the hint should include a solution plan and pseudo code. For Other Reasoning, the hint should be a high-level plan, including suggestions of multiple “experiments” or general thought paths one can consider (e.g., a reminder that there might not be a solution). Refer to the contributor specification document for more details. REMEMBER: if the prompt is not solvable, then the hint should direct the model to the realization that it's unsolvable.

Solution

Hint for Solving the Problem:

The problem is solvable using modular arithmetic and properties of polynomial division. Here’s the step-by-step approach:

Key Information Extraction:
- $p(a) \equiv 0 \pmod{5}$ means $a$ is a root of $p(x)$ modulo 5.
- The division of $p(x)$ by $(x - a)(x + 2)$ gives a remainder $3x + 4$ .
- We need to find the remainder when $p(x)$ is divided by $x + 2$ .
Using Remainder Properties:
- By the division algorithm, $p(x) = (x - a)(x + 2)q(x) + (3x + 4)$ , where $q(x)$ is the quotient.
- Setting $x = -2$ , we find $p(-2)$ as it gives the remainder when dividing $p(x)$ by $x + 2$ .
Simplify and Solve:
- Substitute $x = -2$ into $p(x)$ using the expression $p(x) = (x - a)(x + 2)q(x) + 3x + 4$ .
- Use the given congruence $p(a) \equiv 0 \pmod{5}$ to establish relationships and determine $p(-2)$ .
Substitute Specific Values:
- Substitute into the modular condition to simplify computations. After solving, determine the final remainder modulo 5.

Through these steps, the answer simplifies to $\boxed{-2}$ .

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Math Problem Analysis

Mathematical Concepts

Polynomials
Modular Arithmetic
Remainder Theorem
Polynomial Division

Formulas

p(x) = (x - a)(x + 2)q(x) + (3x + 4)
p(a) ≡ 0 (mod 5)

Theorems

Remainder Theorem
Division Algorithm for Polynomials

Suitable Grade Level

Grades 10-12

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