To solve the expression \(\frac{3b^2 + 33b}{11b}\), we can begin by simplifying it step-by-step.
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Factor Out Common Terms: We can factor out a common term in the numerator. The numerator \(3b^2 + 33b\) has a common factor of \(3b\): \[ 3b^2 + 33b = 3b(b + 11) \]
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Rewrite the Expression: Substituting the factored form back into the expression gives us: \[ \frac{3b(b + 11)}{11b} \]
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Cancel Common Terms: We can cancel \(b\) from the numerator and the denominator, assuming \(b \neq 0\): \[ = \frac{3(b + 11)}{11} \]
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Distribute and Final Simplification: Now distribute \(3\) in the numerator: \[ = \frac{3b + 33}{11} \]
Now, let’s represent this in terms of the answer choices given:
- The expression we derived is: \[ \frac{3b + 33}{11} \]
Now we can see that this expression matches choice \(D\): \[ D. \frac{3b + 3}{11} \]
However, upon inspecting carefully, it seems that I made a mistake in interpreting the matches. The expression is actually: \[ \frac{3b + 33}{11} \neq \frac{3b + 3}{11} \]
Thus, the final answer should be: \[ \frac{3b + 33}{11} = \frac{3b}{11} + \frac{33}{11} = \frac{3b}{11} + 3 \]
Therefore, the correct answer is: \[ A. \frac{3b}{11} + 3 \]
So, the answer is: A. \( \frac{3b}{11} + 3 \)
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