Use synthetic division to find the quotient and the remainder. \( \left(x^{5}+x^{3}-x^{2}\right) \div(x-4) \)

To use synthetic division to find the quotient and the remainder of \( (x^{5}+x^{3}-x^{2}) \div (x-4) \), we start by setting up synthetic division with \( 4 \) (the root of \( x-4=0 \)). 1. Write down the coefficients of \( x^{5} + 0x^{4} + x^{3} - x^{2} + 0x + 0 \), which are \( 1, 0, 1, -1, 0, 0 \). 2. Set up the synthetic division: ``` 4 | 1 0 1 -1 0 0 | 4 16 68 272 ------------------------- 1 4 17 67 272 272 ``` 3. Now, perform the division step by step: - Bring down the \( 1 \). - Multiply \( 4 \) by \( 1 \) to get \( 4 \), and add to \( 0 \) to get \( 4 \). - Multiply \( 4 \) by \( 4 \) to get \( 16 \), and add to \( 1 \) to get \( 17 \). - Multiply \( 4 \) by \( 17 \) to get \( 68 \), and add to \( -1 \) to get \( 67 \). - Multiply \( 4 \) by \( 67 \) to get \( 268 \), and add to \( 0 \) to get \( 268 \). - Multiply \( 4 \) by \( 268 \) to get \( 1072 \), and add to \( 0 \) to get \( 1072 \). The final result is the quotient and remainder: - Quotient: \( x^{4} + 4x^{3} + 17x^{2} + 67x + 268 \) - Remainder: \( 1072 \) So, the result of \( (x^5 + x^3 - x^2) \div (x-4) \) is: \[ x^{4} + 4x^{3} + 17x^{2} + 67x + 268 \quad \text{with a remainder of } \quad 1072. \]