Complete the synthetic division problem below

what is the quotient in polynomail form

- 1 | 2        7       5

○ A. 5

○ B. 9

C. 3

○ D. 7

Let's solve the synthetic division problem to find the quotient in polynomial form.

The synthetic division setup is as follows:

- The divisor is \( x + 1 \), which implies the root is \( -1 \).
- The dividend is represented by the coefficients \( 2 \) (for \( x^2 \)), \( 7 \) (for \( x \)), and \( 5 \) (constant term).

### Synthetic Division Process

1. **Bring down the first coefficient**: This is \( 2 \).

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

2. **Multiply** the root \(-1\) by the first coefficient \(2\) and add the result to the next coefficient \(7\):

   \[
   -1 \times 2 = -2, \quad 7 + (-2) = 5
   \]

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

3. **Multiply** the root \(-1\) by the second coefficient \(5\) and add the result to the next coefficient \(5\):

   \[
   -1 \times 5 = -5, \quad 5 + (-5) = 0
   \]

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

### Result

- The quotient polynomial is \( 2x + 5 \) (the last row without the remainder).

- The remainder is \( 0 \), so the division is exact.

Thus, the quotient in polynomial form is \( 2x + 5 \).

### Conclusion

The given options are not polynomial forms, but if the multiple-choice question was asking for a coefficient or value from the quotient:

- The constant term of the quotient is **5**, which corresponds to option **A**.

So the answer is **A. 5**.

Would you like to see more details or have any questions?

Here are 8 related questions you might explore:

1. How do you perform synthetic division for a cubic polynomial?
2. What is the remainder theorem in polynomial division?
3. How does synthetic division differ from long division in polynomials?
4. What happens when you perform synthetic division with a non-zero remainder?
5. How do you identify the quotient and remainder from synthetic division?
6. What role does the root play in synthetic division?
7. How can you verify the result of synthetic division?
8. Can synthetic division be used for all types of divisors?

**Tip:** Remember, synthetic division is much quicker than long division but only works when the divisor is of the form \(x - c\).

Explore how to complete a synthetic division problem step-by-step and find the quotient polynomial. This problem involves polynomial division concepts and the remainder theorem. Suitable for high school students.

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