https://file.aiquickdraw.com/mathbot/file/17265335226442rzqeozq.jpg

The problem from the image is asking to divide the following polynomial using long division:

\[
\frac{3x^3 - 4x^2 - 3}{x^2 + 5x + 1}
\]

Let's go through the steps of long division:

### Step 1: Divide the leading terms
The leading term of the dividend is \( 3x^3 \), and the leading term of the divisor is \( x^2 \). Divide these terms:

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

### Step 2: Multiply the divisor by the quotient
Now, multiply the entire divisor \( x^2 + 5x + 1 \) by \( 3x \):

\[
3x(x^2 + 5x + 1) = 3x^3 + 15x^2 + 3x
\]

### Step 3: Subtract
Now subtract \( 3x^3 + 15x^2 + 3x \) from the original dividend \( 3x^3 - 4x^2 - 3 \):

\[
(3x^3 - 4x^2 - 3) - (3x^3 + 15x^2 + 3x)
\]

This simplifies to:

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

### Step 4: Repeat the process
Next, divide the new leading term \( -19x^2 \) by the leading term of the divisor \( x^2 \):

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

### Step 5: Multiply and subtract again
Multiply \( -19 \) by the divisor \( x^2 + 5x + 1 \):

\[
-19(x^2 + 5x + 1) = -19x^2 - 95x - 19
\]

Now subtract:

\[
(-19x^2 - 3x - 3) - (-19x^2 - 95x - 19)
\]

This simplifies to:

\[
92x + 16
\]

### Step 6: Conclusion
The remainder is \( 92x + 16 \), so the final result of the division is:

\[
3x - 19 + \frac{92x + 16}{x^2 + 5x + 1}
\]

Would you like further details on any of the steps?

### Here are some related questions:
1. How does synthetic division differ from long division?
2. Can this polynomial be factored for easier division?
3. What are the possible ways to verify the result of polynomial division?
4. How does polynomial long division apply to rational functions?
5. What are the applications of polynomial division in calculus?

### Tip:
Always check your work by multiplying the quotient by the divisor and adding the remainder to ensure it matches the original polynomial.

Divide the polynomial (3x^3 - 4x^2 - 3) by (x^2 + 5x + 1) using long division.

Step-by-step guide to solving the polynomial long division problem 3x^3 - 4x^2 - 3 divided by x^2 + 5x + 1. Learn key algebra concepts and formulas used in this AP Precalculus problem. Suitable for 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