\left(x^{3}+13x^{2}+42x+54\right)\div\left(x+9\right)

To divide \(\left(x^{3}+13x^{2}+42x+54\right)\) by \(\left(x+9\right)\), we can use polynomial long division.

Here’s the process step by step:

1. **Divide the first term of the dividend by the first term of the divisor:**

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

2. **Multiply the entire divisor by the result from step 1:**

   \[
   x^2 \cdot (x + 9) = x^3 + 9x^2
   \]

3. **Subtract the result from the original polynomial:**

   \[
   (x^3 + 13x^2 + 42x + 54) - (x^3 + 9x^2) = 4x^2 + 42x + 54
   \]

4. **Repeat the process with the new polynomial:**

   - Divide the first term of the new polynomial by the first term of the divisor:
     \[
     \frac{4x^2}{x} = 4x
     \]
   - Multiply the entire divisor by this result:
     \[
     4x \cdot (x + 9) = 4x^2 + 36x
     \]
   - Subtract this from the new polynomial:
     \[
     (4x^2 + 42x + 54) - (4x^2 + 36x) = 6x + 54
     \]

5. **Repeat the process one more time:**

   - Divide the first term by the first term of the divisor:
     \[
     \frac{6x}{x} = 6
     \]
   - Multiply the entire divisor by this result:
     \[
     6 \cdot (x + 9) = 6x + 54
     \]
   - Subtract:
     \[
     (6x + 54) - (6x + 54) = 0
     \]

Since the remainder is 0, the quotient is:

\[
\boxed{x^2 + 4x + 6}
\]

Would you like any more details on the steps, or do you have any other questions?

---

Here are five related questions you might find interesting:

1. How would you perform polynomial long division when there is a non-zero remainder?
2. What are synthetic division and when is it used instead of long division?
3. Can you factorize the polynomial \(x^3 + 13x^2 + 42x + 54\) using the quotient \(x^2 + 4x + 6\)?
4. How do you check if a polynomial is divisible by another polynomial?
5. What is the Remainder Theorem and how does it relate to polynomial division?

**Tip:** When dividing polynomials, always arrange both the dividend and divisor in descending order of their powers to simplify the process.

Learn how to divide the polynomial x^3 + 13x^2 + 42x + 54 by (x + 9) using polynomial long division. This detailed step-by-step solution covers polynomial division concepts and techniques, suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation