Use the long division method to find the result when 9, x, cubed, plus, 12, x, squared, plus, 10, x, plus, 49x
3
+12x
2
+10x+4 is divided by 3, x, plus, 23x

Let's clarify the given problem step by step. It seems you're asking for the long division of two polynomials.

The dividend appears to be:
\[
9x^3 + 12x^2 + 10x + 49
\]

The divisor seems to be:
\[
3x + 23
\]

We will use polynomial long division to divide the dividend by the divisor.

### Step-by-Step Long Division:

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

   \[
   \frac{9x^3}{3x} = 3x^2
   \]

   So, the first term in the quotient is \(3x^2\).

2. **Multiply the entire divisor by \(3x^2\):**

   \[
   (3x + 23) \cdot 3x^2 = 9x^3 + 69x^2
   \]

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

   \[
   (9x^3 + 12x^2 + 10x + 49) - (9x^3 + 69x^2)
   \]

   This gives us:

   \[
   -57x^2 + 10x + 49
   \]

4. **Divide the first term of the new polynomial by the first term of the divisor:**

   \[
   \frac{-57x^2}{3x} = -19x
   \]

   So, the next term in the quotient is \(-19x\).

5. **Multiply the entire divisor by \(-19x\):**

   \[
   (3x + 23) \cdot (-19x) = -57x^2 - 437x
   \]

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

   \[
   (-57x^2 + 10x + 49) - (-57x^2 - 437x)
   \]

   This gives us:

   \[
   447x + 49
   \]

7. **Divide the first term of the new polynomial by the first term of the divisor:**

   \[
   \frac{447x}{3x} = 149
   \]

   So, the next term in the quotient is \(149\).

8. **Multiply the entire divisor by \(149\):**

   \[
   (3x + 23) \cdot 149 = 447x + 3427
   \]

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

   \[
   (447x + 49) - (447x + 3427)
   \]

   This gives us:

   \[
   -3378
   \]

Thus, the remainder is \(-3378\), and the quotient is:

\[
3x^2 - 19x + 149
\]

So, the final result is:

\[
\frac{9x^3 + 12x^2 + 10x + 49}{3x + 23} = 3x^2 - 19x + 149 \text{ with a remainder of } -3378.
\]

Would you like more details or explanations about any of the steps? Here are 5 related questions you might find interesting:

1. How is synthetic division different from long division in polynomials?
2. Can polynomial division always be simplified to no remainder?
3. What are the applications of polynomial division in calculus?
4. How can we verify the quotient and remainder are correct?
5. How does polynomial long division apply to real-world problems?

**Tip:** Always check the degrees of terms to ensure you perform each division step accurately during long division!

Use the long division method to find the result when 9x^3 + 12x^2 + 10x + 49 is divided by 3x + 23.

Learn how to use polynomial long division to divide 9x^3 + 12x^2 + 10x + 49 by 3x + 23. This problem explores polynomial division in algebra, providing a detailed step-by-step solution. Suitable for students in 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