Write the quotient and remainder when we divide :
`(5x^(3) - 12x^(2) + 12x + 13)` by `(x^(2) - 3x + 4)`

Answer

To find the quotient and remainder when dividing the polynomial \(5x^3 - 12x^2 + 12x + 13\) by \(x^2 - 3x + 4\), we will use polynomial long division. Here are the steps:

### Step 1: Setup the division
We will divide \(5x^3 - 12x^2 + 12x + 13\) by \(x^2 - 3x + 4\).

### Step 2: Divide the leading terms
Divide the leading term of the dividend \(5x^3\) by the leading term of the divisor \(x^2\):
\[ \frac{5x^3}{x^2} = 5x \]
This gives us the first term of the quotient.

### Step 3: Multiply and subtract
Now, multiply \(5x\) by the entire divisor \(x^2 - 3x + 4\):
\[ 5x \cdot (x^2 - 3x + 4) = 5x^3 - 15x^2 + 20x \]
Now, subtract this from the original polynomial:
\[ (5x^3 - 12x^2 + 12x + 13) - (5x^3 - 15x^2 + 20x) \]
This simplifies to:
\[ (-12x^2 + 15x^2) + (12x - 20x) + 13 = 3x^2 - 8x + 13 \]

### Step 4: Repeat the process
Now, we will divide the new polynomial \(3x^2 - 8x + 13\) by \(x^2 - 3x + 4\).

Divide the leading term \(3x^2\) by \(x^2\):
\[ \frac{3x^2}{x^2} = 3 \]
This gives us the next term of the quotient.

### Step 5: Multiply and subtract again
Multiply \(3\) by the entire divisor \(x^2 - 3x + 4\):
\[ 3 \cdot (x^2 - 3x + 4) = 3x^2 - 9x + 12 \]
Now, subtract this from \(3x^2 - 8x + 13\):
\[ (3x^2 - 8x + 13) - (3x^2 - 9x + 12) \]
This simplifies to:
\[ (-8x + 9x) + (13 - 12) = x + 1 \]

### Step 6: Conclusion
Now, we can see that we cannot divide further because the degree of the remainder \(x + 1\) is less than the degree of the divisor \(x^2 - 3x + 4\).

Thus, the quotient is:
\[ 5x + 3 \]
And the remainder is:
\[ x + 1 \]

### Final Answer:
- Quotient: \(5x + 3\)
- Remainder: \(x + 1\)

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