Which expression is equivalent to the expression below? \frac{6c^2 + 3c}{-4c + 2} \frac{2c + 1}{4c – 2} \frac{3c(2c – 1)}{2c + 1} \frac{-3c(2c + 1)^2}{4(2c – 1)^2} 3c -3c
The Correct Answer and Explanation is:
We are given a complex rational expression and need to simplify it: 6c2+3c−4c+22c+14c−2\frac{\frac{6c^2 + 3c}{-4c + 2}}{\frac{2c + 1}{4c – 2}}
Step 1: Simplify the inner expressions
Numerator: 6c2+3c−4c+2\frac{6c^2 + 3c}{-4c + 2}
Factor the numerator and denominator:
- 6c2+3c=3c(2c+1)6c^2 + 3c = 3c(2c + 1)
- −4c+2=−2(2c−1)-4c + 2 = -2(2c – 1)
So the numerator becomes: 3c(2c+1)−2(2c−1)\frac{3c(2c + 1)}{-2(2c – 1)}
Denominator: 2c+14c−2\frac{2c + 1}{4c – 2}
Factor the denominator:
- 4c−2=2(2c−1)4c – 2 = 2(2c – 1)
So the expression becomes: 2c+12(2c−1)\frac{2c + 1}{2(2c – 1)}
Step 2: Combine the complex fraction
Now write the full expression: 3c(2c+1)−2(2c−1)2c+12(2c−1)\frac{\frac{3c(2c + 1)}{-2(2c – 1)}}{\frac{2c + 1}{2(2c – 1)}}
Dividing by a fraction is the same as multiplying by its reciprocal: 3c(2c+1)−2(2c−1)⋅2(2c−1)2c+1\frac{3c(2c + 1)}{-2(2c – 1)} \cdot \frac{2(2c – 1)}{2c + 1}
Cancel out common terms:
- 2c+12c + 1 cancels
- 2c−12c – 1 cancels
- 22 cancels
What remains is: 3c⋅1−1⋅1=−3c\frac{3c \cdot 1}{-1 \cdot 1} = -3c
Final Answer:
−3c\boxed{-3c}
Explanation
The given problem involves simplifying a complex rational expression. A complex rational expression is a fraction where the numerator or denominator (or both) is also a fraction. To simplify, we begin by factoring each part of the expression to make cancellation easier.
The numerator of the complex fraction is 6c2+3c−4c+2\frac{6c^2 + 3c}{-4c + 2}. Factoring the numerator gives us 3c(2c+1)3c(2c + 1), and factoring the denominator gives −2(2c−1)-2(2c – 1). This transforms the numerator into 3c(2c+1)−2(2c−1)\frac{3c(2c + 1)}{-2(2c – 1)}.
The denominator of the complex fraction is 2c+14c−2\frac{2c + 1}{4c – 2}. Factoring the denominator results in 2(2c−1)2(2c – 1), giving 2c+12(2c−1)\frac{2c + 1}{2(2c – 1)}.
Now we divide the two rational expressions. Dividing by a fraction is equivalent to multiplying by its reciprocal. So we multiply the numerator by the reciprocal of the denominator, resulting in: 3c(2c+1)−2(2c−1)⋅2(2c−1)2c+1\frac{3c(2c + 1)}{-2(2c – 1)} \cdot \frac{2(2c – 1)}{2c + 1}
We cancel out all common terms: 2c+12c + 1, 2c−12c – 1, and 2. We’re left with: −3c-3c
This confirms the correct answer is:
−3c\boxed{-3c}.
