Math Problem Statement

Use the remainder theorem to find P(-3) for P(x) = -x^3 - 4x^2 - 3. Specifically, give the quotient and the remainder for the associated division and the value of P(-3).

Solution

To solve this, we use the remainder theorem, which states that if a polynomial P(x)P(x) is divided by xcx - c, the remainder is P(c)P(c). Here, we want P(3)P(-3), so we'll evaluate the polynomial P(x)=x34x23P(x) = -x^3 - 4x^2 - 3 at x=3x = -3.

Step 1: Substitute x=3x = -3 into P(x)P(x)

P(3)=(3)34(3)23P(-3) = -(-3)^3 - 4(-3)^2 - 3

Step 2: Compute each term

  1. (3)3=27(-3)^3 = -27, so (3)3=27-(-3)^3 = 27.
  2. (3)2=9(-3)^2 = 9, so 4(3)2=36-4(-3)^2 = -36.
  3. The constant term remains 3-3.

Step 3: Add up the results

P(3)=27363=12P(-3) = 27 - 36 - 3 = -12

Thus, the remainder is 12-12, and P(3)=12P(-3) = -12.


Step 4: Quotient

For this question, the quotient isn't explicitly calculated here, but you can use synthetic or polynomial long division if required.

Would you like a walkthrough of synthetic division or long division for the quotient?


Let me know if you need further explanation! Here are 5 related questions for deeper understanding:

  1. How do we perform synthetic division for P(x)=x34x23P(x) = -x^3 - 4x^2 - 3 divided by x+3x + 3?
  2. Can the Remainder Theorem help verify roots of a polynomial?
  3. What is the general relationship between the remainder theorem and factor theorem?
  4. How do we use the quotient to reconstruct the original polynomial?
  5. Why does the remainder theorem work?

Tip: The Remainder Theorem is especially useful for checking possible roots or simplifying polynomial evaluations!

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Math Problem Analysis

Mathematical Concepts

Polynomials
Remainder Theorem
Synthetic Substitution

Formulas

Remainder Theorem: P(c) = remainder when P(x) is divided by (x - c)

Theorems

Remainder Theorem

Suitable Grade Level

Grades 9-12