Question

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Use synthetic division to find the zeroes of the function f(x)=x3+x2+4x+4
.

Answer 1

To find the zeroes of the function \( f(x) = x^3 + x^2 + 4x + 4 \) using synthetic division, we will first need to guess a possible rational root using the Rational Root Theorem and then perform synthetic division.

### Step 1: Identify potential rational roots

The Rational Root Theorem states that potential roots of a polynomial \( f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \) can be given by the factors of the constant term divided by the factors of the leading coefficient.

In this case:
- The constant term \( a_0 = 4 \): its factors are \( \pm 1, \pm 2, \pm 4 \).
- The leading coefficient \( a_3 = 1 \): its factors are \( \pm 1 \).

Thus, possible rational roots are \( \pm 1, \pm 2, \pm 4 \).

### Step 2: Test possible roots using synthetic division

Let’s test \( x = -2 \):

Set up synthetic division:

\[
\begin{array}{r|rrrr}
-2 & 1 & 1 & 4 & 4 \\
& & -2 & 2 & -12 \\
\hline
& 1 & -1 & 6 & -8 \\
\end{array}
\]

The bottom row shows the coefficients of the quotient polynomial and the remainder:
- Quotient: \( x^2 - x + 6 \)
- Remainder: \( -8 \)

Since the remainder is not zero, \( x = -2 \) is not a root.

Next, let’s test \( x = -1 \):

Set up synthetic division:

\[
\begin{array}{r|rrrr}
-1 & 1 & 1 & 4 & 4 \\
& & -1 & 0 & -4 \\
\hline
& 1 & 0 & 4 & 0 \\
\end{array}
\]

The bottom row shows the coefficients of the quotient polynomial and the remainder:
- Quotient: \( x^2 + 0x + 4 = x^2 + 4 \)
- Remainder: \( 0 \)

Since the remainder is zero, we have found that \( x = -1 \) is a root.

### Step 3: Factor the polynomial

From our synthetic division, we can factor \( f(x) \) as:

\[
f(x) = (x + 1)(x^2 + 4)
\]

### Step 4: Find the remaining roots

Next, we need to find the roots of \( x^2 + 4 = 0 \):

\[
x^2 + 4 = 0 \implies x^2 = -4 \implies x = \pm 2i
\]

### Step 5: State the zeroes

The zeroes of the function \( f(x) = x^3 + x^2 + 4x + 4 \) are:

\[
x = -1, \quad x = 2i, \quad x = -2i
\]

### Summary of Roots
- The real root is \( x = -1 \).
- The complex roots are \( x = 2i \) and \( x = -2i \).