`y = (3x ^(2) +7)/(x-3)`
Which of the following expressions is equivalent to y ?

Options

A. `3x + 9 - (20)/(x-3)`
B. `3x +9 + (34)/(x-3)`
C. `3x + 43`
D. `3x ^(2) + (9)/(x -3)`

Answer

To find an equivalent expression for \( y = \frac{3x^2 + 7}{x - 3} \), we will use polynomial long division. Here are the steps:

### Step 1: Set up the long division
We will divide \( 3x^2 + 7 \) by \( x - 3 \).

### Step 2: Divide the leading terms
Divide the leading term of the dividend \( 3x^2 \) by the leading term of the divisor \( x \):
\[ \frac{3x^2}{x} = 3x \]
So, the first term of our quotient is \( 3x \).

### Step 3: Multiply and subtract
Multiply \( 3x \) by the entire divisor \( x - 3 \):
\[ 3x \cdot (x - 3) = 3x^2 - 9x \]
Now, subtract this from the original polynomial:
\[ (3x^2 + 7) - (3x^2 - 9x) = 0 + 9x + 7 = 9x + 7 \]

### Step 4: Repeat the process
Now, we need to divide \( 9x \) by \( x \):
\[ \frac{9x}{x} = 9 \]
So, the next term of our quotient is \( 9 \).

### Step 5: Multiply and subtract again
Multiply \( 9 \) by the divisor \( x - 3 \):
\[ 9 \cdot (x - 3) = 9x - 27 \]
Now, subtract this from \( 9x + 7 \):
\[ (9x + 7) - (9x - 27) = 0 + 7 + 27 = 34 \]

### Step 6: Write the final expression
Now we have completed the division, and we can write the result as:
\[ 3x^2 + 7 = (x - 3)(3x + 9) + 34 \]
Dividing both sides by \( x - 3 \) gives us:
\[ \frac{3x^2 + 7}{x - 3} = 3x + 9 + \frac{34}{x - 3} \]

### Final Answer
Thus, the equivalent expression for \( y \) is:
\[ y = 3x + 9 + \frac{34}{x - 3} \]

### Conclusion
The correct option from the given choices is **b) \( 3x + 9 + \frac{34}{x - 3} \)**.

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