Given that (x+1) is a factor of f(x)=x^3+9x^2-12x-20 fully factorise this function

To fully factorize the function \( f(x) = x^3 + 9x^2 - 12x - 20 \) given that \( (x+1) \) is a factor, follow these steps:

1. **Verify \( (x+1) \) is a factor**:
   Since \( (x+1) \) is a factor, \( x = -1 \) should be a root of \( f(x) \).

   Evaluate \( f(-1) \):
   \[
   f(-1) = (-1)^3 + 9(-1)^2 - 12(-1) - 20 = -1 + 9 + 12 - 20 = 0
   \]
   Thus, \( (x+1) \) is indeed a factor of \( f(x) \).

2. **Perform polynomial division**:
   Divide \( f(x) \) by \( (x+1) \) using synthetic division or long division.

   **Synthetic Division**:
   \[
   \begin{array}{r|rrrr}
    -1 & 1 & 9 & -12 & -20 \\
      &   & -1 & -8 & 20 \\
    \hline
      & 1 & 8 & -20 & 0 \\
   \end{array}
   \]
   So, \( f(x) \) divided by \( (x+1) \) gives the quotient \( x^2 + 8x - 20 \).

3. **Factor the quotient**:
   We now need to factor \( x^2 + 8x - 20 \). To do this, find the roots using the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   where \( a = 1 \), \( b = 8 \), and \( c = -20 \).

   \[
   x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot (-20)}}{2 \cdot 1}
   = \frac{-8 \pm \sqrt{64 + 80}}{2}
   = \frac{-8 \pm \sqrt{144}}{2}
   = \frac{-8 \pm 12}{2}
   \]
   So, the roots are:
   \[
   x = \frac{-8 + 12}{2} = 2 \quad \text{and} \quad x = \frac{-8 - 12}{2} = -10
   \]

   Therefore, \( x^2 + 8x - 20 \) factors as \( (x - 2)(x + 10) \).

4. **Combine the factors**:
   Since \( f(x) \) can be written as \( (x+1) \) times the quotient, and the quotient is \( (x - 2)(x + 10) \), the full factorization is:
   \[
   f(x) = (x+1)(x-2)(x+10)
   \]

So, the fully factorized form of \( f(x) \) is:
\[
f(x) = (x+1)(x-2)(x+10)
\]

Would you like more details or have any questions?

Here are some questions you might find useful:
1. How do you perform synthetic division in more detail?
2. Can you show how to verify if a polynomial factor is correct?
3. What are other methods to factorize polynomials?
4. How do you find the roots of a quadratic equation using the quadratic formula?
5. Can you explain polynomial long division?

**Tip:** When factoring polynomials, always start by checking for obvious roots or factors and use division to simplify the polynomial step by step.

Learn how to fully factorize the polynomial f(x)=x^3+9x^2-12x-20 given that (x+1) is a factor. Follow a detailed step-by-step solution including polynomial division and quadratic equation roots. Suitable for advanced high school students (Grades 10-12).

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