\( 1 \leftarrow \quad \) Use synthetic division to find the quotient and remainder when \( x^{5}-9 x^{3}+2 x \) is divided by \( x+1 \) Quotient: Remainder:

To perform synthetic division of \( x^{5}-9x^{3}+2x \) by \( x+1 \) (which corresponds to the root \( -1 \)), you first list the coefficients of the polynomial: 1. For \( x^5 \), the coefficient is 1. 2. For \( x^4 \), the coefficient is 0 (since \( x^4 \) is missing). 3. For \( x^3 \), the coefficient is -9. 4. For \( x^2 \), the coefficient is 0 (since \( x^2 \) is missing). 5. For \( x^1 \), the coefficient is 2. 6. For the constant term \( x^0 \), the coefficient is 0 (since there is no constant term). So the coefficients are: \( 1, 0, -9, 0, 2, 0 \). Now, setting up the synthetic division with -1: ``` -1 | 1 0 -9 0 2 0 | -1 1 8 -8 6 -------------------------- 1 -1 -8 8 -6 6 ``` Reading the results from the bottom row, the coefficients for the quotient correspond to \( x^4, x^3, x^2, x^1 \), so the quotient is \( x^4 - x^3 - 8x^2 + 8x - 6 \) and the remainder is 6. Quotient: \( x^4 - x^3 - 8x^2 + 8x - 6 \) Remainder: 6