solve with formula and solution (5x³+ 11x² + 26x +26) ÷ (5×+6) synthetic division

To solve the polynomial division \((5x³ + 11x² + 26x + 26) \div (5x + 6)\) using synthetic division, follow these steps:

### Step 1: Setup Synthetic Division
First, recognize that the divisor \(5x + 6\) can be written as \(5(x + \frac{6}{5})\). For synthetic division, we use the zero of the divisor, which is \(x = -\frac{6}{5}\).

### Step 2: Coefficients of the Dividend
List the coefficients of the polynomial \(5x³ + 11x² + 26x + 26\):

- Coefficients: \(5\), \(11\), \(26\), \(26\)

### Step 3: Perform Synthetic Division
1. Bring down the leading coefficient \(5\).
2. Multiply \(5\) by \(-\frac{6}{5}\) and add to the next coefficient.
3. Repeat the process for each coefficient.

#### Process:

\[
\begin{array}{r|rrrr}
-\frac{6}{5} & 5 & 11 & 26 & 26 \\
  &   & -6 & -6 & -36 \\
\hline
  & 5 & 5 & 20 & -10 \\
\end{array}
\]

### Step 4: Interpret the Results
- The quotient polynomial is \(5x² + 5x + 20\).
- The remainder is \(-10\).

### Final Answer
The division yields:

\[
\frac{5x³ + 11x² + 26x + 26}{5x + 6} = 5x² + 5x + 20 - \frac{10}{5x + 6}
\]

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**Related Questions:**

1. How is synthetic division different from long division in polynomials?
2. Can synthetic division be used with divisors of higher degrees?
3. What is the significance of the remainder in polynomial division?
4. How can you factor a polynomial using the results from synthetic division?
5. What are some real-world applications of polynomial division?

**Tip:** Always ensure that the divisor is a linear polynomial (degree 1) before using synthetic division.

solve with formula and solution (5x³+ 11x² + 26x +26) ÷ (5x+6) synthetic division

Explore the step-by-step solution for dividing the polynomial 5x³ + 11x² + 26x + 26 by 5x + 6 using synthetic division. This detailed guide covers key algebraic concepts, including synthetic division and the remainder theorem. Ideal for students in Grades 10-12.

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