use long devision to devide p(x) = 3x^4-5x^3+2x^2-x+4 by d(x) = x^2-2x+3. Express the result in the form p(x) = d(x) x q(x) +r(x)

To divide \( p(x) = 3x^4 - 5x^3 + 2x^2 - x + 4 \) by \( d(x) = x^2 - 2x + 3 \) using long division, we follow these steps:

1. **Set up the long division**:  
   We divide \( 3x^4 - 5x^3 + 2x^2 - x + 4 \) by \( x^2 - 2x + 3 \).

2. **First division**:  
   Divide the leading term of \( p(x) \) by the leading term of \( d(x) \):  
   \[
   \frac{3x^4}{x^2} = 3x^2
   \]
   Multiply \( 3x^2 \) by \( x^2 - 2x + 3 \):
   \[
   3x^2 \cdot (x^2 - 2x + 3) = 3x^4 - 6x^3 + 9x^2
   \]
   Subtract from \( p(x) \):
   \[
   (3x^4 - 5x^3 + 2x^2 - x + 4) - (3x^4 - 6x^3 + 9x^2) = (0x^4) + (x^3) - (7x^2) - x + 4
   \]
   The new polynomial is \( x^3 - 7x^2 - x + 4 \).

3. **Second division**:  
   Divide the leading term of the new polynomial by the leading term of \( d(x) \):  
   \[
   \frac{x^3}{x^2} = x
   \]
   Multiply \( x \) by \( x^2 - 2x + 3 \):
   \[
   x \cdot (x^2 - 2x + 3) = x^3 - 2x^2 + 3x
   \]
   Subtract from the new polynomial:
   \[
   (x^3 - 7x^2 - x + 4) - (x^3 - 2x^2 + 3x) = (0x^3) - 5x^2 - 4x + 4
   \]
   The new polynomial is \( -5x^2 - 4x + 4 \).

4. **Third division**:  
   Divide the leading term of the new polynomial by the leading term of \( d(x) \):  
   \[
   \frac{-5x^2}{x^2} = -5
   \]
   Multiply \( -5 \) by \( x^2 - 2x + 3 \):
   \[
   -5 \cdot (x^2 - 2x + 3) = -5x^2 + 10x - 15
   \]
   Subtract from the new polynomial:
   \[
   (-5x^2 - 4x + 4) - (-5x^2 + 10x - 15) = (0x^2) - 14x + 19
   \]
   The new polynomial is \( -14x + 19 \).

5. **Conclusion**:  
   The quotient is \( q(x) = 3x^2 + x - 5 \) and the remainder is \( r(x) = -14x + 19 \).

Thus, we can express \( p(x) \) as:
\[
p(x) = d(x) \cdot q(x) + r(x)
\]
which gives:
\[
3x^4 - 5x^3 + 2x^2 - x + 4 = (x^2 - 2x + 3)(3x^2 + x - 5) + (-14x + 19)
\]

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Would you like more details on any of the steps?  
Here are 5 related questions:

1. How do you perform polynomial long division in more complex cases?
2. What happens if the remainder is zero in polynomial long division?
3. Can synthetic division be used for this problem instead of long division?
4. How can polynomial division be applied in calculus?
5. How can we find the roots of the quotient or remainder?

**Tip**: Always check your final answer by multiplying the quotient and divisor and adding the remainder.

Use long division to divide p(x) = 3x^4 - 5x^3 + 2x^2 - x + 4 by d(x) = x^2 - 2x + 3. Express the result in the form p(x) = d(x) * q(x) + r(x).

Learn how to perform polynomial long division to divide 3x^4 - 5x^3 + 2x^2 - x + 4 by x^2 - 2x + 3. This detailed step-by-step solution covers polynomial division, resulting in a quotient and remainder. Suitable for students in Grades 10-12.

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