divide 𝑃(𝑥) = 3𝑥4 − 6𝑥3 +3𝑥 −6 por Q(x)= (𝑥2 − 𝑥 +1).

To divide the polynomial \( P(x) = 3x^4 - 6x^3 + 3x - 6 \) by \( Q(x) = x^2 - x + 1 \), we will perform polynomial long division.

### Steps:

1. **Divide the first term of \( P(x) \) by the first term of \( Q(x) \):**

   \[
   \frac{3x^4}{x^2} = 3x^2
   \]

2. **Multiply \( 3x^2 \) by \( Q(x) \) (i.e., \( x^2 - x + 1 \)):**

   \[
   3x^2(x^2 - x + 1) = 3x^4 - 3x^3 + 3x^2
   \]

3. **Subtract the result from \( P(x) \):**

   \[
   (3x^4 - 6x^3 + 3x - 6) - (3x^4 - 3x^3 + 3x^2) = -3x^3 - 3x^2 + 3x - 6
   \]

4. **Divide the first term of the new polynomial by the first term of \( Q(x) \):**

   \[
   \frac{-3x^3}{x^2} = -3x
   \]

5. **Multiply \( -3x \) by \( Q(x) \):**

   \[
   -3x(x^2 - x + 1) = -3x^3 + 3x^2 - 3x
   \]

6. **Subtract this result from the current polynomial:**

   \[
   (-3x^3 - 3x^2 + 3x - 6) - (-3x^3 + 3x^2 - 3x) = -6x^2 + 6x - 6
   \]

7. **Divide the first term of the new polynomial by the first term of \( Q(x) \):**

   \[
   \frac{-6x^2}{x^2} = -6
   \]

8. **Multiply \( -6 \) by \( Q(x) \):**

   \[
   -6(x^2 - x + 1) = -6x^2 + 6x - 6
   \]

9. **Subtract this result from the current polynomial:**

   \[
   (-6x^2 + 6x - 6) - (-6x^2 + 6x - 6) = 0
   \]

The remainder is 0, and the quotient is:

\[
\boxed{3x^2 - 3x - 6}
\]

This means:

\[
\frac{P(x)}{Q(x)} = 3x^2 - 3x - 6
\]

Would you like further explanation or have any questions?

Here are some related questions:

1. How do we interpret the result of dividing polynomials?
2. Can synthetic division be applied to divide polynomials like this?
3. How can we check the result of polynomial division?
4. What is the remainder theorem for polynomials?
5. How does polynomial long division relate to factoring?

**Tip:** Always verify your polynomial division by multiplying the quotient by the divisor to check if you get the original polynomial.

divide P(x) = 3x^4 - 6x^3 + 3x - 6 by Q(x) = x^2 - x + 1.

This detailed guide walks through dividing the polynomial P(x) = 3x^4 - 6x^3 + 3x - 6 by Q(x) = x^2 - x + 1 using polynomial long division. Learn step-by-step techniques for solving this algebraic problem, suitable for high school students (Grades 10-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation