To find the quotient and remainder of the polynomial \(\frac{x^3 - 3x^2 + 5}{x + 1}\) using synthetic division, follow these steps:
1. Identify the coefficients of the polynomial and the root of the divisor:
- The polynomial is \(x^3 - 3x^2 + 5\). The coefficients are: \(1, -3, 0, 5\).
- The divisor is \(x + 1\), which means the root is \(-1\) (set \(x + 1 = 0\) and solve for \(x\)).
2. Set up the synthetic division table:
- Write down the coefficients of the polynomial: [tex]\[1, -3, 0, 5\][/tex]
- Write the root of the divisor to the left: [tex]\[-1\][/tex]
3. Perform the synthetic division:
- Bring down the leading coefficient (1) to the bottom row: [tex]\[1\][/tex]
- Multiply the root (-1) by the value just written in the bottom row (1) and write it under the second coefficient (-3): [tex]\[-1\][/tex]
- Add the values in the second column: \(-3 + (-1) = -4\)
- Multiply the root (-1) by the value just written in the bottom row (-4) and write it under the third coefficient (0):[tex]\[4\][/tex]
- Add the values in the third column: \(0 + 4 = 4\)
- Multiply the root (-1) by the value just written in the bottom row (4) and write it under the fourth coefficient (5):[tex]\[-4\][/tex]
- Add the values in the fourth column: \(5 + (-4) = 1\)
4. Interpret the results:
- The last row of values is the quotient and remainder: \([1, -4, 4]\) with a remainder of \(1\)
- The quotient corresponds to the coefficients of \(x^2 - 4x + 4\)
- The remainder is the constant term \(1\)
Thus, the quotient is \(1x^2 - 4x + 4\) (which simplifies to \(x^2 - 4x + 4\)), and the remainder is \(1\).
### Final Answer:
- The quotient is \(\boxed{x^2 - 4x + 4}\)
- The remainder is [tex]\(\boxed{1}\)[/tex]
