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 \( x = -2i \).
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