To find the additional linear factors of the polynomial \( x^3 - 6x^2 + 11x - 6 \) given that \( x - 3 \) is a factor, we can perform polynomial long division to divide the polynomial by \( x - 3 \).
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Set up the long division with \( x^3 - 6x^2 + 11x - 6 \) divided by \( x - 3 \).
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Divide the leading term:
- \( x^3 \div x = x^2 \)
- Multiply \( x^2 \) by \( x - 3 \) to get \( x^3 - 3x^2 \).
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Subtract this from the original polynomial: \[ (x^3 - 6x^2 + 11x - 6) - (x^3 - 3x^2) = -3x^2 + 11x - 6 \]
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Repeat the process:
- \( -3x^2 \div x = -3x \)
- Multiply \( -3x \) by \( x - 3 \) to get \( -3x^2 + 9x \).
- Subtract: \[ (-3x^2 + 11x - 6) - (-3x^2 + 9x) = 2x - 6 \]
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Repeat again:
- \( 2x \div x = 2 \)
- Multiply \( 2 \) by \( x - 3 \) to get \( 2x - 6 \).
- Subtract: \[ (2x - 6) - (2x - 6) = 0 \]
The result of the division is \( x^2 - 3x + 2 \).
Now factor \( x^2 - 3x + 2 \):
- This can be factored as \( (x - 1)(x - 2) \).
Thus, the complete factorization of \( x^3 - 6x^2 + 11x - 6 \) is: \[ (x - 3)(x - 1)(x - 2) \]
The additional linear factors of the polynomial are \( (x - 1)(x - 2) \), or in the response format:
(x−2)(x−1).
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