https://file.aiquickdraw.com/mathbot/file/1727271203231821fn9s1.jpg

The problem in the image involves dividing the third-degree polynomial \(8x^3 + 20x^2 - 12x\) by \(x + 3\) using long division.

Let's solve this step by step using polynomial long division.

We are dividing:
\[
\frac{8x^3 + 20x^2 - 12x}{x + 3}
\]

### Step 1: Divide the leading terms
Divide \(8x^3\) by \(x\), which gives \(8x^2\).

### Step 2: Multiply and subtract
Multiply \(8x^2\) by \(x + 3\), resulting in:
\[
8x^2(x + 3) = 8x^3 + 24x^2
\]
Now subtract \(8x^3 + 24x^2\) from the original polynomial:
\[
(8x^3 + 20x^2 - 12x) - (8x^3 + 24x^2) = -4x^2 - 12x
\]

### Step 3: Divide the next leading term
Divide \(-4x^2\) by \(x\), which gives \(-4x\).

### Step 4: Multiply and subtract
Multiply \(-4x\) by \(x + 3\):
\[
-4x(x + 3) = -4x^2 - 12x
\]
Now subtract \(-4x^2 - 12x\) from \(-4x^2 - 12x\):
\[
(-4x^2 - 12x) - (-4x^2 - 12x) = 0
\]

Since there are no more terms to divide, the quotient is:
\[
8x^2 - 4x
\]

Thus, the correct answer is **Option #2**: \(8x^2 - 4x\).

Would you like more details or have any further questions?

Here are 5 related questions for practice:
1. What is synthetic division and how does it differ from long division?
2. How do you divide a polynomial by a binomial using synthetic division?
3. What is the remainder theorem, and how does it apply in polynomial division?
4. Can the long division method be used for dividing higher-degree polynomials?
5. How do you verify the result of polynomial division?

**Tip**: Always check your result by multiplying the quotient by the divisor to see if you get back the original polynomial.

Divide the third-degree polynomial, 8x^3 + 20x^2 - 12x, by the first-degree polynomial, x + 3, using long division. What is the result?

Learn how to divide the polynomial 8x^3 + 20x^2 - 12x by x + 3 using the long division method. This problem illustrates key algebraic concepts such as polynomial division and the division algorithm for polynomials. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation