Write the quotient and remainder when we divide :
`(14x^(2) - 53x + 45)` by (7x - 9)

Answer

To find the quotient and remainder when dividing \( 14x^2 - 53x + 45 \) by \( 7x - 9 \), we can use polynomial long division. Here are the steps:

### Step 1: Set up the division
We will divide \( 14x^2 - 53x + 45 \) by \( 7x - 9 \).

### Step 2: Divide the leading terms
Divide the leading term of the dividend \( 14x^2 \) by the leading term of the divisor \( 7x \):
\[ \frac{14x^2}{7x} = 2x \]
This gives us the first term of the quotient.

**Hint:** Always divide the leading term of the dividend by the leading term of the divisor.

### Step 3: Multiply and subtract
Now, multiply \( 2x \) by the entire divisor \( 7x - 9 \):
\[ 2x \cdot (7x - 9) = 14x^2 - 18x \]
Subtract this from the original polynomial:
\[ (14x^2 - 53x + 45) - (14x^2 - 18x) = -53x + 18x + 45 = -35x + 45 \]

**Hint:** Remember to change the signs when subtracting.

### Step 4: Repeat the process
Now, we need to divide the new leading term \( -35x \) by \( 7x \):
\[ \frac{-35x}{7x} = -5 \]
This is the next term of the quotient.

**Hint:** Continue the process by focusing on the new leading term.

### Step 5: Multiply and subtract again
Multiply \( -5 \) by the entire divisor \( 7x - 9 \):
\[ -5 \cdot (7x - 9) = -35x + 45 \]
Subtract this from the current polynomial:
\[ (-35x + 45) - (-35x + 45) = 0 \]

**Hint:** Always ensure to align like terms when subtracting.

### Conclusion
At this point, we have completed the division. The quotient is:
\[ \text{Quotient} = 2x - 5 \]
And the remainder is:
\[ \text{Remainder} = 0 \]

### Final Answer
- Quotient: \( 2x - 5 \)
- Remainder: \( 0 \)