Divide using synthetic division. {:[(x^(5)+3x^(4)+2x^(3)+4x^(2)+5x-5)/(x+2)],[(x^(5)+3x^(4)+2x^(3)+4x^(2)+5x-5)/(x+2)= _____]:} (Simplify your answer. Use integers or fractions for any numbers in the expression. Do not factor.)

Question

Divide using synthetic division.  {:[(x^(5)+3x^(4)+2x^(3)+4x^(2)+5x-5)/(x+2)],[(x^(5)+3x^(4)+2x^(3)+4x^(2)+5x-5)/(x+2)= _____]:} (Simplify your answer. Use integers or fractions for any numbers in the expression. Do not factor.)

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Set up division: To perform synthetic division, we first need to set up the division. The divisor is $x + 2$, so the number we use for synthetic division is the opposite of the constant term of the divisor, which is -2.Write coefficients: Write down the coefficients of the dividend polynomial $x^5 + 3x^4 + 2x^3 + 4x^2 + 5x - 5$, which are 1, 3, 2, 4, 5, and -5.Bring down leading coefficient: Begin the synthetic division process by bringing down the leading coefficient (which is 1) to the bottom row.Multiply and add: Multiply the number just written on the bottom row by -2 (the number we use for synthetic division) and write the result in the next column of the second row, under the second coefficient (which is 3). Calculation: $1 \times -2 = -2$.Repeat process for third coefficient: Add the numbers in the second column to get the new second coefficient in the bottom row. Calculation: $3 + (-2) = 1$.Continue with fourth coefficient: Repeat the multiplication and addition process for the third coefficient. Multiply the new second coefficient (which is 1) by -2 and add it to the third coefficient (which is 2). Calculation: $1 \times -2 = -2$, then $2 + (-2) = 0$.Proceed with fifth coefficient: Continue the process for the fourth coefficient. Multiply the new third coefficient (which is 0) by -2 and add it to the fourth coefficient (which is 4). Calculation: $0 \times -2 = 0$, then $4 + 0 = 4$.Address last coefficient: Proceed with the fifth coefficient. Multiply the new fourth coefficient (which is 4) by -2 and add it to the fifth coefficient (which is 5). Calculation: $4 \times -2 = -8$, then $5 + (-8) = -3$.Determine quotient and remainder: Finally, address the last coefficient. Multiply the new fifth coefficient (which is -3) by -2 and add it to the last coefficient (which is -5). Calculation: $-3 \times -2 = 6$, then $-5 + 6 = 1$.The bottom row now represents the coefficients of the quotient polynomial. The remainder is the last number on the bottom row. The quotient polynomial is $x^4 + x^3 + 4x - 3$ with a remainder of 1.

Divide using synthetic division. $\newline$ $\begin{array}{l} \frac{x^{5}+3 x^{4}+2 x^{3}+4 x^{2}+5 x-5}{x+2} \\ \frac{x^{5}+3 x^{4}+2 x^{3}+4 x^{2}+5 x-5}{x+2}= \_\_\_\_\_ \\ \end{array}$ $\newline$ (Simplify your answer. Use integers or fractions for any numbers in the expression. Do not factor.)

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