Write the quotient and remainder when we divide :
`(x^(3) - 6x^(2) + 11x - 6)` by `(x^(2) - 5x + 6)`

Answer

To find the quotient and remainder when dividing \( x^3 - 6x^2 + 11x - 6 \) by \( x^2 - 5x + 6 \), we can use polynomial long division. Here are the steps:

### Step 1: Set up the division
We will divide \( x^3 - 6x^2 + 11x - 6 \) (the dividend) by \( x^2 - 5x + 6 \) (the divisor).

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

### Step 3: Multiply and subtract
Now, multiply the entire divisor \( x^2 - 5x + 6 \) by \( x \):
\[ x(x^2 - 5x + 6) = x^3 - 5x^2 + 6x \]
Now, subtract this result from the original polynomial:
\[ (x^3 - 6x^2 + 11x - 6) - (x^3 - 5x^2 + 6x) \]
This simplifies to:
\[ (-6x^2 + 5x^2) + (11x - 6x) - 6 = -x^2 + 5x - 6 \]

### Step 4: Repeat the process
Now we need to divide \( -x^2 + 5x - 6 \) by \( x^2 - 5x + 6 \).

Divide the leading term \( -x^2 \) by \( x^2 \):
\[ \frac{-x^2}{x^2} = -1 \]
This \( -1 \) is the next term of our quotient.

### Step 5: Multiply and subtract again
Multiply the entire divisor by \( -1 \):
\[ -1(x^2 - 5x + 6) = -x^2 + 5x - 6 \]
Now, subtract this from \( -x^2 + 5x - 6 \):
\[ (-x^2 + 5x - 6) - (-x^2 + 5x - 6) = 0 \]

### Step 6: Conclusion
Since we have reached 0, the division is complete.

Thus, the quotient is:
\[ \text{Quotient} = x - 1 \]
And the remainder is:
\[ \text{Remainder} = 0 \]

### Final Answer:
Quotient: \( x - 1 \)
Remainder: \( 0 \)

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