divide x^3-9x^2-22x+120 by x+4 using synthetic division. then with that solution, find factored form of original equation including (x+4)

To divide \(x^3 - 9x^2 - 22x + 120\) by \(x + 4\) using synthetic division, we proceed as follows:

1. **Set up the synthetic division**: 
   - The divisor is \(x + 4\), so we use \(-4\).
   - The coefficients of the polynomial \(x^3 - 9x^2 - 22x + 120\) are \(1, -9, -22, 120\).

2. **Perform the synthetic division**:

```
-4 | 1  -9  -22  120
   |     -4   52  -120
   -----------------
     1 -13   30    0
```

Here's the step-by-step breakdown:
- Bring down the 1.
- Multiply \(-4\) by 1 and write \(-4\) under \(-9\), then add to get \(-13\).
- Multiply \(-4\) by \(-13\) and write \(52\) under \(-22\), then add to get \(30\).
- Multiply \(-4\) by \(30\) and write \(-120\) under \(120\), then add to get \(0\).

So, the quotient is \(x^2 - 13x + 30\) and the remainder is \(0\).

3. **Find the factored form of the original equation**:
   - We know \(x + 4\) is one factor.
   - We need to factor \(x^2 - 13x + 30\).

To factor \(x^2 - 13x + 30\):
- We look for two numbers that multiply to 30 and add to -13. These numbers are -3 and -10.

Thus, \(x^2 - 13x + 30\) factors to \((x - 3)(x - 10)\).

4. **Combine all the factors**:
   - The original polynomial \(x^3 - 9x^2 - 22x + 120\) can be factored as \((x + 4)(x - 3)(x - 10)\).

So, the factored form of the polynomial \(x^3 - 9x^2 - 22x + 120\) is \((x + 4)(x - 3)(x - 10)\).

Would you like more details or have any questions about this process?

Here are some related questions for further practice:
1. What is synthetic division, and how is it different from long division?
2. How do you determine which value to use for synthetic division?
3. Can you perform synthetic division with a divisor other than a linear polynomial?
4. What are the steps to factor a quadratic equation?
5. How do you verify the factored form of a polynomial?
6. What should you do if the remainder is not zero in synthetic division?
7. How can synthetic division be used to find polynomial roots?
8. How do you handle synthetic division when coefficients are fractions or decimals?

**Tip:** Always verify your final factorization by multiplying the factors back together to ensure you obtain the original polynomial.

Learn how to divide x^3 - 9x^2 - 22x + 120 by x + 4 using synthetic division and find the factored form of the original polynomial. This problem covers polynomial division, synthetic division technique, and factoring polynomials. Suitable for high school students.

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