Use synthetic division to find the result when 2, x, cubed, minus, 7, x, squared, plus, 12, x, minus, 72x 
3
 −7x 
2
 +12x−7 is divided by x, minus, 1x−1.

Synthetic division is a simplified form of polynomial division, particularly useful for dividing a polynomial by a linear term of the form \( x - c \). In this case, we're dividing the polynomial \( 2x^3 - 7x^2 + 12x - 7 \) by \( x - 1 \).

### Step 1: Set Up for Synthetic Division

The coefficients of the polynomial \( 2x^3 - 7x^2 + 12x - 7 \) are \( 2, -7, 12, \) and \(-7\). We use \( c = 1 \) from the divisor \( x - 1 \).

\[
\begin{array}{r|rrrr}
1 & 2 & -7 & 12 & -7 \\
  &   &    &    &   \\
\end{array}
\]

### Step 2: Perform the Synthetic Division

1. **Bring down the leading coefficient** \( 2 \).

\[
\begin{array}{r|rrrr}
1 & 2 & -7 & 12 & -7 \\
  &   &    &    &   \\
  & 2 &    &    &   \\
\end{array}
\]

2. **Multiply** the value of \( c \) (which is \( 1 \)) by the number just written below the line (\( 2 \)) and write the result under the next coefficient.

\[
\begin{array}{r|rrrr}
1 & 2 & -7 & 12 & -7 \\
  &   & 2  &    &   \\
  & 2 &    &    &   \\
\end{array}
\]

3. **Add** the column.

\[
\begin{array}{r|rrrr}
1 & 2 & -7 & 12 & -7 \\
  &   & 2  &    &   \\
  & 2 & -5 &    &   \\
\end{array}
\]

4. **Repeat** the process: Multiply the result (\(-5\)) by \( 1 \) and write it under the next coefficient.

\[
\begin{array}{r|rrrr}
1 & 2 & -7 & 12 & -7 \\
  &   & 2  & -5 &   \\
  & 2 & -5 &    &   \\
\end{array}
\]

5. **Add** the column.

\[
\begin{array}{r|rrrr}
1 & 2 & -7 & 12 & -7 \\
  &   & 2  & -5 & 7 \\
  & 2 & -5 & 7  &   \\
\end{array}
\]

6. **Repeat** once more: Multiply the result (\( 7 \)) by \( 1 \) and write it under the next coefficient.

\[
\begin{array}{r|rrrr}
1 & 2 & -7 & 12 & -7 \\
  &   & 2  & -5 & 7 \\
  & 2 & -5 & 7  &   \\
\end{array}
\]

7. **Add** the last column.

\[
\begin{array}{r|rrrr}
1 & 2 & -7 & 12 & -7 \\
  &   & 2  & -5 & 7 \\
  & 2 & -5 & 7  & 0 \\
\end{array}
\]

### Step 3: Interpret the Results

The bottom row, excluding the last number, gives the coefficients of the quotient polynomial. The last number is the remainder. Therefore, the quotient is:

\[
2x^2 - 5x + 7
\]

And the remainder is \( 0 \).

### Conclusion

When \( 2x^3 - 7x^2 + 12x - 7 \) is divided by \( x - 1 \), the result is:

\[
2x^2 - 5x + 7
\]

Do you want details or have any questions?

Here are some related questions for further exploration:
1. What is synthetic division, and when is it useful?
2. How does synthetic division differ from long division in polynomials?
3. Can synthetic division be used with divisors other than linear polynomials?
4. How can synthetic division help in finding the roots of a polynomial?
5. What are some potential errors to watch out for when using synthetic division?
6. How do you use synthetic division to test for potential factors of a polynomial?
7. Can synthetic division be extended to polynomials with complex coefficients?
8. What is the Remainder Theorem, and how is it related to synthetic division?

**Tip:** Remember, synthetic division is particularly efficient for polynomials divided by linear terms of the form \( x - c \).

Learn how to use synthetic division to divide the polynomial 2x^3 - 7x^2 + 12x - 7 by x - 1. Follow a step-by-step solution with detailed explanations and examples. Suitable for advanced high school students and anyone studying polynomial division.

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