Which of the following is equivalent to the expressions `(7x-4)/(x+9)`?

Options

A. `7-(4)/(x+9)`
B. `7-(67)/(x+9)`
C. `7-(4x)/(9)`
D. `(7-4x)/(9)`

Answer

To solve the problem of finding which expression is equivalent to \((7x - 4)/(x + 9)\), we can use polynomial long division. Here’s a step-by-step solution:

### Step 1: Set up the long division
We want to divide \(7x - 4\) by \(x + 9\).

### Step 2: Divide the leading terms
Divide the leading term of the dividend \(7x\) by the leading term of the divisor \(x\):
\[ \frac{7x}{x} = 7 \]
So, the first term of our quotient is \(7\).

### Step 3: Multiply and subtract
Now, multiply the entire divisor \(x + 9\) by \(7\):
\[ 7(x + 9) = 7x + 63 \]
Next, subtract this from the original dividend:
\[ (7x - 4) - (7x + 63) \]
Distributing the negative sign gives:
\[ 7x - 4 - 7x - 63 = -4 - 63 = -67 \]

### Step 4: Write the result
Now we can express our division result using the division algorithm:
\[ \frac{7x - 4}{x + 9} = 7 + \frac{-67}{x + 9} \]
This can be rewritten as:
\[ \frac{7x - 4}{x + 9} = 7 - \frac{67}{x + 9} \]

### Conclusion
Thus, the expression \((7x - 4)/(x + 9)\) is equivalent to \(7 - \frac{67}{x + 9}\).

### Final Answer
The correct option from the given choices is **B: \(7 - \frac{67}{x + 9}\)**.

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