Perform the indicated division and write your answers in the form P(x)/D(x)=Q(x)+R(x)/D(x) shown in the following example

(5x²-17x-15)➗ (x-4)

(Use the long division method and the synthetic method)

Let's solve the division \((5x^2 - 17x - 15) \div (x - 4)\) using both the **long division** method and the **synthetic division** method.

### 1. **Long Division Method**:

We are dividing \( P(x) = 5x^2 - 17x - 15 \) by \( D(x) = x - 4 \).

**Step 1:** Divide the leading term of \( P(x) \) by the leading term of \( D(x) \).
\[
\frac{5x^2}{x} = 5x
\]
So, the first term of the quotient \( Q(x) \) is \( 5x \).

**Step 2:** Multiply \( 5x \) by \( (x - 4) \) and subtract from \( P(x) \).
\[
5x(x - 4) = 5x^2 - 20x
\]
Now subtract:
\[
(5x^2 - 17x - 15) - (5x^2 - 20x) = (0x^2 + 3x - 15)
\]

**Step 3:** Divide the leading term \( 3x \) by \( x \):
\[
\frac{3x}{x} = 3
\]
So, the next term in the quotient is \( 3 \).

**Step 4:** Multiply \( 3 \) by \( (x - 4) \) and subtract:
\[
3(x - 4) = 3x - 12
\]
Now subtract:
\[
(3x - 15) - (3x - 12) = -3
\]

So, the remainder is \( R(x) = -3 \).

**Final Answer in Long Division Form**:
\[
\frac{5x^2 - 17x - 15}{x - 4} = 5x + 3 - \frac{3}{x - 4}
\]

### 2. **Synthetic Division Method**:

In synthetic division, we use the root of the divisor \( x - 4 \), which is \( 4 \), and divide \( P(x) = 5x^2 - 17x - 15 \) by \( x - 4 \).

We use the coefficients of \( P(x) \): \( 5, -17, -15 \).

Set up the synthetic division:

\[
\begin{array}{r|rrrr}
4 & 5 & -17 & -15 \\
  &   & 20  & 12  \\
\hline
  & 5 & 3   & -3  \\
\end{array}
\]

- Bring down the \( 5 \).
- Multiply \( 5 \times 4 = 20 \), and add to \( -17 \): \( -17 + 20 = 3 \).
- Multiply \( 3 \times 4 = 12 \), and add to \( -15 \): \( -15 + 12 = -3 \).

The quotient is \( 5x + 3 \), and the remainder is \( -3 \).

**Final Answer in Synthetic Division Form**:
\[
\frac{5x^2 - 17x - 15}{x - 4} = 5x + 3 - \frac{3}{x - 4}
\]

### Summary:
The final answer is the same using both methods:
\[
\frac{5x^2 - 17x - 15}{x - 4} = 5x + 3 - \frac{3}{x - 4}
\]

---

Would you like more details or have any questions on either method?

#### Related Questions:
1. How do you handle polynomial division when the divisor is more than a linear binomial?
2. How do you apply the Remainder Theorem to polynomial division?
3. What is the role of the degree of polynomials in division?
4. How does synthetic division change when dividing by \(x + a\)?
5. Can synthetic division be used for all polynomials?

**Tip**: Always check the degree of the remainder to ensure it is less than the degree of the divisor in polynomial division.

Perform the indicated division and write your answers in the form P(x)/D(x)=Q(x)+R(x)/D(x) shown in the following example: (5x²-17x-15) ÷ (x-4) (Use the long division method and the synthetic method).

Learn how to divide the polynomial 5x² - 17x - 15 by x - 4 using both long division and synthetic division methods. This detailed step-by-step solution covers key algebraic concepts, including polynomial division and the Remainder Theorem, suitable for Grades 9-12.

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