To decompose the polynomial \( x^3 + 3x^2 - 4x - 12 \) into its linear factors, given that \( x - 2 \) is a factor, we can use polynomial long division or synthetic division to divide the polynomial by \( x - 2 \).
Let's start with the polynomial division:
- Divide the leading term: Divide the leading term \( x^3 \) by \( x \) (the leading term of \( x - 2 \)). The result is \( x^2 \).
- Multiply: Multiply \( x^2 \) by \( x - 2 \) to get \( x^3 - 2x^2 \).
- Subtract: Subtract \( (x^3 - 2x^2) \) from \( x^3 + 3x^2 \) to get \( (3x^2 + 2x^2) = 5x^2 \).
- Bring Down: Now, bring down the next term, which is \(-4x\), giving \( 5x^2 - 4x \).
- Repeat the division: Divide \( 5x^2 \) by \( x \) to get \( 5x \).
- Multiply: Multiply \( 5x \) by \( x - 2 \) to get \( 5x^2 - 10x \).
- Subtract: Subtract \( (5x^2 - 10x) \) from \( 5x^2 - 4x \) to get \( ( -4x + 10x ) = 6x \).
- Bring down: Bring down the last term, \(-12\), giving \( 6x - 12 \).
- Repeat the division: Divide \( 6x \) by \( x \) to get \( 6 \).
- Multiply: Multiply \( 6 \) by \( x - 2 \) to get \( 6x - 12 \).
- Subtract: Subtract \( 6x - 12 \) from \( 6x - 12 \) to get \( 0 \).
The quotient from the division is: \[ x^2 + 5x + 6 \]
Next, we need to factor \( x^2 + 5x + 6 \) further: \[ x^2 + 5x + 6 = (x + 2)(x + 3) \]
Now we combine everything: \[ x^3 + 3x^2 - 4x - 12 = (x - 2)(x + 2)(x + 3) \]
So the polynomial \( x^3 + 3x^2 - 4x - 12 \) can be decomposed into its linear factors as: \[ (x - 2)(x + 2)(x + 3) \]
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