Divide Using Synthetic Division (2x^5-x^4+3x^2-x+5)/(x-1)

Problem

(2*x^5−x^4+3*x^2−x+5)/(x−1)

Solution

1. Identify the coefficients of the dividend 2*x^5−x^4+0*x^3+3*x^2−x+5 Note that a 0 is used for the missing x^3 term.

2. Determine the divisor constant c from the expression x−c Since the divisor is x−1 c=1

3. Set up the synthetic division table by writing 1 to the left and the coefficients $2, -1, 0, 3, -1, 5$ in a row.

4. Bring down the first coefficient 2 to the bottom row.

5. Multiply the divisor 1 by the value in the bottom row 2 and write the result 2 under the next coefficient −1

6. Add the column values −1+2=1 and write the result in the bottom row.

7. Repeat the multiply-and-add process for the remaining columns:

1⋅1=1⇒0+1=1

1⋅1=1⇒3+1=4

1⋅4=4⇒−1+4=3

1⋅3=3⇒5+3=8

8. Interpret the bottom row $2, 1, 1, 4, 3, 8$ as the coefficients of the quotient (starting one degree lower than the dividend) and the remainder.

Final Answer

(2*x^5−x^4+3*x^2−x+5)/(x−1)=2*x^4+x^3+x^2+4*x+3+8/(x−1)

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