Use long division or synthetic division in finding the quotient: 2x^3 -x^2 - 22x - 25 ÷ x-3*

2 points

2x^2 + 9x +14 + 71/x-3

2x^2 - 5x - 7 + 46/x-3

2x^2 + 5x - 7 -46/x-3

2x^2 - 9x +14 - 71/x-3

To solve this division problem, let's use **synthetic division** to divide \( 2x^3 - x^2 - 22x - 25 \) by \( x - 3 \).

### Step 1: Set up synthetic division.

Since we're dividing by \( x - 3 \), we'll use \( 3 \) as the number for synthetic division. The coefficients of the polynomial \( 2x^3 - x^2 - 22x - 25 \) are:

- \( 2 \) (coefficient of \( x^3 \))
- \( -1 \) (coefficient of \( x^2 \))
- \( -22 \) (coefficient of \( x \))
- \( -25 \) (constant)

So the setup for synthetic division is as follows:

\[
\begin{array}{r|rrrr}
3 & 2 & -1 & -22 & -25 \\
\end{array}
\]

### Step 2: Perform synthetic division.

1. Bring down the first coefficient (2) unchanged.
2. Multiply 2 by 3, getting 6. Write this under the next coefficient (-1).
3. Add \( -1 + 6 = 5 \).
4. Multiply 5 by 3, getting 15. Write this under the next coefficient (-22).
5. Add \( -22 + 15 = -7 \).
6. Multiply -7 by 3, getting -21. Write this under the next constant (-25).
7. Add \( -25 + (-21) = -46 \).

The synthetic division table looks like this:

\[
\begin{array}{r|rrrr}
3 & 2 & -1 & -22 & -25 \\
  &   & 6  & 15  & -21 \\
\hline
  & 2 & 5  & -7  & -46 \\
\end{array}
\]

### Step 3: Write the quotient and remainder.

The quotient is obtained from the top row, so it is:

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

The remainder is \( -46 \), so the full result of the division is:

\[
2x^2 + 5x - 7 - \frac{46}{x - 3}
\]

### Final Answer:
The correct result is:

\[
\boxed{2x^2 + 5x - 7 - \frac{46}{x - 3}}
\]

Would you like further explanation or have any other questions?

### 5 Related Questions:
1. How does synthetic division differ from long division of polynomials?
2. Can synthetic division be used for divisors other than \( x - c \) (linear divisors)?
3. What are the advantages of using synthetic division over long division?
4. How do you determine if a division yields a remainder of zero?
5. Can you verify the division by multiplying the quotient and divisor back together?

### Tip:
Remember that synthetic division can only be used when the divisor is a linear polynomial of the form \( x - c \). For higher-degree divisors, use long division.

Use long division or synthetic division in finding the quotient: 2x^3 -x^2 - 22x - 25 ÷ x-3

Learn how to solve the polynomial division 2x^3 -x^2 - 22x - 25 ÷ x-3 using synthetic division. This detailed step-by-step guide covers algebraic concepts like synthetic division setup and the remainder theorem. Ideal for students in Grades 10-12.

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