Since \( x = 2 \) is a zero of the polynomial \( f(x) = x^3 - 3x^2 - 10x + 24 \), by the Factor Theorem, we know that \( (x - 2) \) is a factor of \( f(x) \).
To factor \( f(x) \), we can use synthetic division to divide \( f(x) \) by \( (x - 2) \).
Perform synthetic division with \( 2 \):
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Write down the coefficients of \( f(x) \): \[ 1, -3, -10, 24 \]
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Set up the synthetic division: \[ \begin{array}{r|rrrr} 2 & 1 & -3 & -10 & 24 \ & & 2 & -2 & -24 \ \hline & 1 & -1 & -12 & 0 \ \end{array} \]
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The bottom row gives the coefficients of the quotient polynomial: \[ 1x^2 - 1x - 12 = x^2 - x - 12 \]
Now we can factor \( x^2 - x - 12 \):
To factor \( x^2 - x - 12 \), we need to find two numbers that multiply to \(-12\) and add to \(-1\). The numbers \(-4\) and \(3\) fit this requirement.
Thus, we can factor \( x^2 - x - 12 \) as follows: \[ x^2 - x - 12 = (x - 4)(x + 3) \]
Now we can write the full factorization of \( f(x) \): \[ f(x) = (x - 2)(x - 4)(x + 3) \]
In conclusion, the full factorization of \( f(x) \) into linear factors is: \[ f(x) = (x - 2)(x - 4)(x + 3) \]