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

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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Precalculus

Divide polynomials using synthetic division

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Q. 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.)

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 $3$ ). $\newline$ 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. $\newline$ 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 $2$ ). $\newline$ 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 $4$ ). $\newline$ 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 $5$ ). $\newline$ 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 $-5$ ). $\newline$ Calculation: $-3 \times -2 = 6$ , then $-5 + 6 = 1$ .
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 $-5$ ). $\newline$ 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. $\newline$ The quotient polynomial is $x^4 + x^3 + 4x - 3$ with a remainder of $1$ .