Bot GPT 4-o mini To determine which of the given options is a factor of the polynomial \( f(x) = 2x^3 - 5x^2 - 68x + 35 \), we can apply the Factor Theorem. According to the Factor Theorem, \( (x - c) \) is a factor of \( f(x) \) if and only if \( f(c) = 0 \).
We will test each option by substituting the corresponding value of \( x \) into \( f(x) \).
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Testing \( (x + 5) \):
- Set \( c = -5 \): \[ f(-5) = 2(-5)^3 - 5(-5)^2 - 68(-5) + 35 \] \[ = 2(-125) - 5(25) + 340 + 35 \] \[ = -250 - 125 + 340 + 35 \] \[ = -250 - 125 + 375 = 0 \]
So \( (x + 5) \) is a factor of \( f(x) \).
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Testing \( (2x + 1) \):
- Set \( c = -\frac{1}{2} \): \[ f\left(-\frac{1}{2}\right) = 2\left(-\frac{1}{2}\right)^3 - 5\left(-\frac{1}{2}\right)^2 - 68\left(-\frac{1}{2}\right) + 35 \] \[ = 2\left(-\frac{1}{8}\right) - 5\left(\frac{1}{4}\right) + 34 + 35 \] \[ = -\frac{1}{4} - \frac{5}{4} + 34 + 35 \] \[ = -\frac{6}{4} + 69 = -1.5 + 69 = 67.5 \neq 0 \]
So \( (2x + 1) \) is not a factor.
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Testing \( (x - 3) \):
- Set \( c = 3 \): \[ f(3) = 2(3)^3 - 5(3)^2 - 68(3) + 35 \] \[ = 2(27) - 5(9) - 204 + 35 \] \[ = 54 - 45 - 204 + 35 \] \[ = 54 - 45 - 204 + 35 = -160 \neq 0 \]
So \( (x - 3) \) is not a factor.
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Testing \( (x + 7) \):
- Set \( c = -7 \): \[ f(-7) = 2(-7)^3 - 5(-7)^2 - 68(-7) + 35 \] \[ = 2(-343) - 5(49) + 476 + 35 \] \[ = -686 - 245 + 476 + 35 \] \[ = -686 - 245 + 511 = -420 \neq 0 \]
So \( (x + 7) \) is not a factor.
In conclusion, the only factor from the provided options is:
\[ \textbf{(x + 5)} \]
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