Division of Polynomials USE 2 METHODS: SYNTHETIC DISION AND LONG DIVISION

Example 1: Alex is stuck on a problem while working on dividing polynomials. Can you help him to obtain the quotient:

(×^4 +2x^3 -13x^2 -38x - 24) = (x-4)

Certainly! Let’s help Alex solve the division of polynomials problem \( \frac{x^4 + 2x^3 - 13x^2 - 38x - 24}{x - 4} \) using both **Synthetic Division** and **Long Division**.

### Step 1: **Synthetic Division**

Synthetic division is a simplified form of polynomial division. Since we are dividing by \( x - 4 \), we'll set \( x = 4 \) as the divisor.

1. **Write down the coefficients** of the polynomial \( x^4 + 2x^3 - 13x^2 - 38x - 24 \). The coefficients are:
   \[
   1 \quad 2 \quad -13 \quad -38 \quad -24
   \]
   
2. **Perform synthetic division** using \( 4 \) as the divisor:

\[
\begin{array}{r|rrrrr}
4 & 1 & 2 & -13 & -38 & -24 \\
  &   & 4 & 24 & 44  & 24 \\
\hline
  & 1 & 6 & 11  & 6   & 0 \\
\end{array}
\]

3. **Result**:
   The quotient is \( x^3 + 6x^2 + 11x + 6 \) and the remainder is \( 0 \). 

Thus, using synthetic division, we find:
\[
\frac{x^4 + 2x^3 - 13x^2 - 38x - 24}{x - 4} = x^3 + 6x^2 + 11x + 6
\]

---

### Step 2: **Long Division**

Now, let’s solve the same problem using long division.

1. **Set up the division**:
   \[
   \frac{x^4 + 2x^3 - 13x^2 - 38x - 24}{x - 4}
   \]

2. **Divide the leading term** \( x^4 \) by \( x \), which gives \( x^3 \).

3. **Multiply** \( x^3 \) by \( x - 4 \), which gives \( x^4 - 4x^3 \).

4. **Subtract** \( (x^4 - 4x^3) \) from \( (x^4 + 2x^3) \), resulting in \( 6x^3 - 13x^2 \).

5. **Divide** \( 6x^3 \) by \( x \), which gives \( 6x^2 \).

6. **Multiply** \( 6x^2 \) by \( x - 4 \), which gives \( 6x^3 - 24x^2 \).

7. **Subtract** \( (6x^3 - 24x^2) \) from \( (6x^3 - 13x^2) \), resulting in \( 11x^2 - 38x \).

8. **Divide** \( 11x^2 \) by \( x \), which gives \( 11x \).

9. **Multiply** \( 11x \) by \( x - 4 \), which gives \( 11x^2 - 44x \).

10. **Subtract** \( (11x^2 - 44x) \) from \( (11x^2 - 38x) \), resulting in \( 6x - 24 \).

11. **Divide** \( 6x \) by \( x \), which gives \( 6 \).

12. **Multiply** \( 6 \) by \( x - 4 \), which gives \( 6x - 24 \).

13. **Subtract** \( (6x - 24) \) from \( (6x - 24) \), resulting in \( 0 \).

Thus, the quotient from long division is also:
\[
x^3 + 6x^2 + 11x + 6
\]

---

### Final Answer:
In both methods, the quotient is:
\[
x^3 + 6x^2 + 11x + 6
\]

Would you like further clarification or additional details on any step?

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### Five Relative Questions:
1. How do you identify the divisor in synthetic division?
2. What should you do if there's a remainder in polynomial division?
3. Why is the leading coefficient important in both division methods?
4. Can synthetic division be used for divisors other than linear polynomials like \( x - 4 \)?
5. What are the advantages of using synthetic division over long division?

---

### Tip:
Always align terms by degree in long division to avoid mistakes when subtracting.

Division of Polynomials USE 2 METHODS: SYNTHETIC DISION AND LONG DIVISION

Example 1: Alex is stuck on a problem while working on dividing polynomials. Can you help him to obtain the quotient:

(x^4 + 2x^3 -13x^2 -38x - 24) ÷ (x-4)?

Learn how to divide polynomials such as (x^4 + 2x^3 - 13x^2 - 38x - 24) by (x - 4) using both Synthetic Division and Long Division. Follow a step-by-step solution that covers key algebraic concepts and theorems such as the Remainder Theorem and the Factor Theorem. Suitable for high school students in Grades 9-12.

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