The expression `(7y-3)/(y+2)` is equivalent to which of the following?

Options

A. `7-(3)/(2y)`
B. `(7-3)/(3y)`
C. `7-(17)/(y+2)`
D. `7-(3)/(y+2)`

Answer

To solve the expression \((7y - 3)/(y + 2)\) and find its equivalent form, we will perform polynomial long division. Here are the steps:

### Step 1: Set up the division
We need to divide \(7y - 3\) by \(y + 2\).

### Step 2: Divide the leading terms
Divide the leading term of the numerator \(7y\) by the leading term of the denominator \(y\):
\[ \frac{7y}{y} = 7 \]
This means that \(7\) is the first term of our quotient.

### Step 3: Multiply and subtract
Now, multiply \(7\) by the entire divisor \(y + 2\):
\[ 7(y + 2) = 7y + 14 \]
Next, subtract this result from the original numerator:
\[ (7y - 3) - (7y + 14) \]
Distributing the negative sign gives:
\[ 7y - 3 - 7y - 14 = -3 - 14 = -17 \]

### Step 4: Write the result
Now we can express the original division as:
\[ \frac{7y - 3}{y + 2} = 7 + \frac{-17}{y + 2} \]
So, the expression is equivalent to:
\[ 7 - \frac{17}{y + 2} \]

### Final Answer
Thus, the expression \((7y - 3)/(y + 2)\) is equivalent to:
\[ 7 - \frac{17}{y + 2} \]