Then why in synthetic division we get 10x² +35x -25

Synthetic division is a method used to divide a polynomial by a binomial of the form \( x - c \). To explain why you get \( 10x^2 + 35x - 25 \) in your synthetic division result, let’s walk through an example. Suppose you are dividing a polynomial \( P(x) = 10x^3 + 15x^2 + 20x - 50 \) by \( x - 1 \). Here's how the synthetic division would proceed:

1. **Set up the division**: Write down the coefficients of \( P(x) \):
   \[
   10, \quad 15, \quad 20, \quad -50
   \]
   The divisor is \( x - 1 \), so \( c = 1 \).

2. **Perform the synthetic division**:

   - Start with the leading coefficient, 10. Bring it down as it is:
     \[
     10
     \]
   - Multiply this 10 by \( c = 1 \) and write the result under the next coefficient:
     \[
     10 \times 1 = 10
     \]
   - Add this result to the next coefficient (15):
     \[
     15 + 10 = 25
     \]
   - Multiply 25 by \( c = 1 \) and write the result under the next coefficient:
     \[
     25 \times 1 = 25
     \]
   - Add this result to the next coefficient (20):
     \[
     20 + 25 = 45
     \]
   - Multiply 45 by \( c = 1 \) and write the result under the next coefficient:
     \[
     45 \times 1 = 45
     \]
   - Add this result to the last coefficient (-50):
     \[
     -50 + 45 = -5
     \]

   The new coefficients from top to bottom are:
   \[
   10, \quad 25, \quad 45, \quad -5
   \]

   So the quotient is:
   \[
   10x^2 + 25x + 45 \quad \text{with a remainder of } -5
   \]

### Why \( 10x^2 + 35x - 25 \)?

Given that the coefficients 10, 25, 45 don't directly match your expected result \( 10x^2 + 35x - 25 \), it’s likely that a different divisor or coefficients in the original polynomial were used.

1. **If you started with different polynomial**: Perhaps the polynomial was \( P(x) = 10x^3 + 45x^2 + 25x - 25 \) divided by \( x - 1 \), then the synthetic division would give \( 10x^2 + 35x - 25 \) with some remainder.

2. **Other possibilities**: Another scenario is a different \( c \) value or sign changes that led to the specific result.

Would you like more detailed steps on how synthetic division works, or to clarify which exact polynomial and divisor you used?

### Follow-up Questions:
1. What is the specific polynomial and divisor you used for this synthetic division?
2. Would you like to see the complete step-by-step synthetic division for your example?
3. Are there any specific points in synthetic division that are confusing to you?
4. Would you like to understand how to handle remainders in synthetic division?
5. Do you need help with the relationship between synthetic division and the Factor Theorem?

**Tip**: Always double-check the signs and coefficients when performing synthetic division to avoid errors in the final polynomial.

Explore synthetic division with an example, explaining how to divide polynomials using this method. Learn step-by-step with detailed calculations to understand the process. Suitable for advanced high school students.

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