Evaluate the function f(x)=(x^3)+(x^2)−3x+9 for x=−4 using synthetic division

To evaluate the function \( f(x) = x^3 + x^2 - 3x + 9 \) at \( x = -4 \) using synthetic division, we will follow these steps:

### Set Up the Synthetic Division:

1. **Write the coefficients** of the polynomial \( f(x) = x^3 + x^2 - 3x + 9 \). The coefficients are \( 1, 1, -3, 9 \).

2. **Set up the synthetic division** by writing \( -4 \) (the value of \( x \)) to the left of a vertical bar and the coefficients \( 1, 1, -3, 9 \) to the right.

### Perform the Synthetic Division:

We perform synthetic division by carrying out the following calculations:

\[
\begin{array}{r|rrrr}
-4 & 1 & 1 & -3 & 9 \\
   &   & -4 & 12 & -36 \\
\hline
   & 1 & -3 & 9 & -27 \\
\end{array}
\]

Let's go through this step-by-step:

1. **Bring down** the first coefficient, \(1\).
   
2. **Multiply** this coefficient by \(-4\) (the value of \(x\)) to get \(-4\), and **write** it under the next coefficient \(1\).

3. **Add** the column: \(1 + (-4) = -3\). Write \(-3\) below.

4. **Multiply** \(-3\) by \(-4\) to get \(12\), and **write** it under the next coefficient \(-3\).

5. **Add** the column: \(-3 + 12 = 9\). Write \(9\) below.

6. **Multiply** \(9\) by \(-4\) to get \(-36\), and **write** it under the next coefficient \(9\).

7. **Add** the column: \(9 + (-36) = -27\). Write \(-27\) below.

The last number, \(-27\), is the remainder, which is also the value of \( f(-4) \).

### Conclusion:

The value of the function \( f(x) = x^3 + x^2 - 3x + 9 \) at \( x = -4 \) is \( f(-4) = -27 \).

Would you like more details on synthetic division, or do you have any questions?

Here are five related questions:
1. How do you interpret the remainder in synthetic division when evaluating a polynomial?
2. Can synthetic division be used to find roots of a polynomial? How?
3. What happens when the remainder in synthetic division is zero?
4. How would synthetic division change if evaluating at \( x = 4 \) instead of \( x = -4 \)?
5. How can synthetic division be used to factor polynomials?

**Tip:** Remember that synthetic division is particularly useful for quickly evaluating polynomials at specific values and finding roots when \( x - c \) is a factor of the polynomial.

Learn how to evaluate the function f(x) = x^3 + x^2 - 3x + 9 at x = -4 using synthetic division with a detailed step-by-step solution. This problem covers polynomials and synthetic division, suitable for students in Grades 10-12.

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