To determine which expression is not a factor of the polynomial \(6x^3 - 35x^2 - 69x + 140\), we can perform polynomial long division with each given expression. However, since you didn't provide the specific expressions, I'll explain how to perform the long division and indicate how to identify a non-factor.
Steps for Polynomial Long Division:
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Divide the leading terms: Divide the leading term of the polynomial by the leading term of the divisor.
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Multiply: Multiply the entire divisor by the result obtained in step 1.
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Subtract: Subtract the result from the original polynomial.
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Bring down the next term: If there are more terms, bring down the next term from the polynomial.
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Repeat: Continue the process until all terms have been brought down.
Determining Non-Factors:
- If, after performing the long division, you obtain a remainder of 0, then the divisor is a factor of the polynomial.
- If there is a remainder that is not 0, then the divisor is not a factor of the polynomial.
Example:
Suppose one of the expressions provided is \(x - 2\).
Let's perform long division on \(6x^3 - 35x^2 - 69x + 140\) by \(x - 2\):
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Divide \(6x^3\) by \(x\): \[ 6x^2 \]
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Multiply \(x - 2\) by \(6x^2\): \[ 6x^3 - 12x^2 \]
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Subtract: \[ (6x^3 - 35x^2) - (6x^3 - 12x^2) = -35x^2 + 12x^2 = -23x^2 \] Bring down the next term \(-69x\): \[ -23x^2 - 69x \]
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Divide \(-23x^2\) by \(x\): \[ -23x \]
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Multiply \(x - 2\) by \(-23x\): \[ -23x^2 + 46x \]
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Subtract: \[ (-23x^2 - 69x) - (-23x^2 + 46x) = -69x - 46x = -115x \] Bring down the \(+140\): \[ -115x + 140 \]
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Divide \(-115x\) by \(x\): \[ -115 \]
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Multiply \(x - 2\) by \(-115\): \[ -115x + 230 \]
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Subtract: \[ (-115x + 140) - (-115x + 230) = 140 - 230 = -90 \]
Since we ended up with a remainder of \(-90\), \(x - 2\) is not a factor of \(6x^3 - 35x^2 - 69x + 140\).
You can repeat this process for each expression provided. Let me know if you have specific expressions listed so we can evaluate them directly.
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