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Let's start by solving the problems one by one using synthetic division. 

### 1. \( (8x^4 - 14x^2 + x^2 - 12x + 12) \div (x - 2) \)

**Step 1**: Write down the coefficients of the polynomial:
- \(8x^4 + 0x^3 - 14x^2 + x - 12 + 12\)
The coefficients are: \[8, 0, -14, 1, -12, 12\]

Now, synthetic division with \(x - 2\) (so the value is \(2\)).

\[
\begin{array}{r|rrrrrr}
2 & 8 & 0 & -14 & 1 & -12 & 12 \\
  &   & 16 & 32  & 36 & 74 &  124 \\
\hline
  & 8 & 16 & 18  & 37 & 62 & 136
\end{array}
\]

So, the quotient is \(8x^3 + 16x^2 + 18x + 37\), and the remainder is 136.

### 2. \( (n^3 + 15n^2 + 47n - 38) \div (n + 6) \)

Coefficients: \([1, 15, 47, -38]\)

Using \(n + 6\), synthetic division with \(-6\):

\[
\begin{array}{r|rrrr}
-6 & 1 & 15 & 47 & -38 \\
   &   & -6 & -54 & 42  \\
\hline
   & 1 & 9  & -7  & 4
\end{array}
\]

The quotient is \(n^2 + 9n - 7\), and the remainder is 4.

### 3. \( (10r^2 + 50r - 60) \div (10r - 10) \)

Divide coefficients: \([10, 50, -60]\)

Using synthetic division with \(r = 1\):

\[
\begin{array}{r|rrr}
1 & 10 & 50 & -60 \\
  &    & 10 & 60  \\
\hline
  & 10 & 60 & 0
\end{array}
\]

The quotient is \(10r + 60\), and the remainder is 0.

Would you like a more detailed explanation of any step? Here are five related questions to expand your understanding:

1. What are the conditions for using synthetic division instead of long division for polynomials?
2. How do you handle missing terms when setting up synthetic division?
3. What happens when the divisor is not of the form \(x - a\)?
4. How does synthetic division help find roots of polynomials?
5. Can synthetic division be applied to non-monomial divisors? How?

**Tip**: Always ensure you have accounted for any missing terms in the polynomial by inserting coefficients of zero when setting up synthetic division.

Divide the given polynomial expressions using synthetic division.

This tutorial provides step-by-step solutions to polynomial division problems using synthetic division. Learn how to apply synthetic division to divide polynomials, identify remainders, and interpret results. Suitable for students in Grades 9-11.

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