Answer :
We want to divide
[tex]$$
x^4 + 15x^3 - 77x^2 + 13x - 36
$$[/tex]
by
[tex]$$
x - 4.
$$[/tex]
Since the divisor is in the form [tex]$x - c$[/tex], we can use synthetic division with [tex]$c=4$[/tex]. Follow these steps:
1. Write down the coefficients of the polynomial:
[tex]$$1,\quad 15,\quad -77,\quad 13,\quad -36.$$[/tex]
2. Draw a line and write the synthetic root [tex]$4$[/tex] to the left. The synthetic division set-up looks like this:
[tex]$$
\begin{array}{cccccc}
\quad & 4 & \quad & \quad & \quad & \\[6mm]
1 & 15 & -77 & 13 & -36 \\
\quad & \underline{\quad} & \quad & \quad & \quad \\
\end{array}
$$[/tex]
3. Bring down the first coefficient (1) below the line:
[tex]$$
\begin{array}{cccccc}
\quad & 4 & \quad & \quad & \quad & \\[6mm]
1 & 15 & -77 & 13 & -36 \\
\quad & & & & \\
\hline
1 & & & &
\end{array}
$$[/tex]
4. Multiply the number just brought down (1) by [tex]$4$[/tex]:
[tex]$$1 \times 4 = 4.$$[/tex]
Write this result below the next coefficient (15):
[tex]$$
\begin{array}{cccccc}
\quad & 4 & \quad & \quad & \quad & \\[6mm]
1 & 15 & -77 & 13 & -36 \\
\quad & 4 & & & \\
\hline
1 & & & &
\end{array}
$$[/tex]
5. Add the second coefficient (15) and the product (4):
[tex]$$15 + 4 = 19.$$[/tex]
Write 19 under the line next to 1.
[tex]$$
\begin{array}{cccccc}
\quad & 4 & \quad & \quad & \quad & \\[6mm]
1 & 15 & -77 & 13 & -36 \\
\quad & 4 & & & \\
\hline
1 & 19 & & &
\end{array}
$$[/tex]
6. Multiply the new entry (19) by [tex]$4$[/tex]:
[tex]$$19 \times 4 = 76.$$[/tex]
Write 76 under the next coefficient (-77):
[tex]$$
\begin{array}{cccccc}
\quad & 4 & \quad & \quad & \quad & \\[6mm]
1 & 15 & -77 & 13 & -36 \\
\quad & 4 & 76 & & \\
\hline
1 & 19 & & &
\end{array}
$$[/tex]
7. Add the third coefficient (-77) and 76:
[tex]$$-77 + 76 = -1.$$[/tex]
Write -1 under the line.
[tex]$$
\begin{array}{cccccc}
\quad & 4 & \quad & \quad & \quad & \\[6mm]
1 & 15 & -77 & 13 & -36 \\
\quad & 4 & 76 & & \\
\hline
1 & 19 & -1 & &
\end{array}
$$[/tex]
8. Multiply -1 by [tex]$4$[/tex]:
[tex]$$-1 \times 4 = -4.$$[/tex]
Write -4 under the fourth coefficient (13):
[tex]$$
\begin{array}{cccccc}
\quad & 4 & \quad & \quad & \quad & \\[6mm]
1 & 15 & -77 & 13 & -36 \\
\quad & 4 & 76 & -4 & \\
\hline
1 & 19 & -1 & &
\end{array}
$$[/tex]
9. Add the fourth coefficient (13) and -4:
[tex]$$13 + (-4) = 9.$$[/tex]
Write 9 under the line.
[tex]$$
\begin{array}{cccccc}
\quad & 4 & \quad & \quad & \quad & \\[6mm]
1 & 15 & -77 & 13 & -36 \\
\quad & 4 & 76 & -4 & \\
\hline
1 & 19 & -1 & 9 &
\end{array}
$$[/tex]
10. Multiply 9 by [tex]$4$[/tex]:
[tex]$$9 \times 4 = 36.$$[/tex]
Write 36 under the fifth coefficient (-36).
[tex]$$
\begin{array}{cccccc}
\quad & 4 & \quad & \quad & \quad & \\[6mm]
1 & 15 & -77 & 13 & -36 \\
\quad & 4 & 76 & -4 & 36 \\
\hline
1 & 19 & -1 & 9 &
\end{array}
$$[/tex]
11. Finally, add the fifth coefficient (-36) and 36:
[tex]$$-36 + 36 = 0.$$[/tex]
Write 0 as the remainder.
[tex]$$
\begin{array}{cccccc}
\quad & 4 & \quad & \quad & \quad & \\[6mm]
1 & 15 & -77 & 13 & -36 \\
\quad & 4 & 76 & -4 & 36 \\
\hline
1 & 19 & -1 & 9 & 0
\end{array}
$$[/tex]
The numbers on the bottom row (except for the last number which is the remainder) are the coefficients of the quotient polynomial. Thus, the quotient is
[tex]$$
x^3 + 19x^2 - x + 9
$$[/tex]
and the remainder is [tex]$0$[/tex].
Therefore, the result is:
[tex]$$
\frac{x^4+15x^3-77x^2+13x-36}{x-4} = x^3+19x^2-x+9.
$$[/tex]
[tex]$$
x^4 + 15x^3 - 77x^2 + 13x - 36
$$[/tex]
by
[tex]$$
x - 4.
$$[/tex]
Since the divisor is in the form [tex]$x - c$[/tex], we can use synthetic division with [tex]$c=4$[/tex]. Follow these steps:
1. Write down the coefficients of the polynomial:
[tex]$$1,\quad 15,\quad -77,\quad 13,\quad -36.$$[/tex]
2. Draw a line and write the synthetic root [tex]$4$[/tex] to the left. The synthetic division set-up looks like this:
[tex]$$
\begin{array}{cccccc}
\quad & 4 & \quad & \quad & \quad & \\[6mm]
1 & 15 & -77 & 13 & -36 \\
\quad & \underline{\quad} & \quad & \quad & \quad \\
\end{array}
$$[/tex]
3. Bring down the first coefficient (1) below the line:
[tex]$$
\begin{array}{cccccc}
\quad & 4 & \quad & \quad & \quad & \\[6mm]
1 & 15 & -77 & 13 & -36 \\
\quad & & & & \\
\hline
1 & & & &
\end{array}
$$[/tex]
4. Multiply the number just brought down (1) by [tex]$4$[/tex]:
[tex]$$1 \times 4 = 4.$$[/tex]
Write this result below the next coefficient (15):
[tex]$$
\begin{array}{cccccc}
\quad & 4 & \quad & \quad & \quad & \\[6mm]
1 & 15 & -77 & 13 & -36 \\
\quad & 4 & & & \\
\hline
1 & & & &
\end{array}
$$[/tex]
5. Add the second coefficient (15) and the product (4):
[tex]$$15 + 4 = 19.$$[/tex]
Write 19 under the line next to 1.
[tex]$$
\begin{array}{cccccc}
\quad & 4 & \quad & \quad & \quad & \\[6mm]
1 & 15 & -77 & 13 & -36 \\
\quad & 4 & & & \\
\hline
1 & 19 & & &
\end{array}
$$[/tex]
6. Multiply the new entry (19) by [tex]$4$[/tex]:
[tex]$$19 \times 4 = 76.$$[/tex]
Write 76 under the next coefficient (-77):
[tex]$$
\begin{array}{cccccc}
\quad & 4 & \quad & \quad & \quad & \\[6mm]
1 & 15 & -77 & 13 & -36 \\
\quad & 4 & 76 & & \\
\hline
1 & 19 & & &
\end{array}
$$[/tex]
7. Add the third coefficient (-77) and 76:
[tex]$$-77 + 76 = -1.$$[/tex]
Write -1 under the line.
[tex]$$
\begin{array}{cccccc}
\quad & 4 & \quad & \quad & \quad & \\[6mm]
1 & 15 & -77 & 13 & -36 \\
\quad & 4 & 76 & & \\
\hline
1 & 19 & -1 & &
\end{array}
$$[/tex]
8. Multiply -1 by [tex]$4$[/tex]:
[tex]$$-1 \times 4 = -4.$$[/tex]
Write -4 under the fourth coefficient (13):
[tex]$$
\begin{array}{cccccc}
\quad & 4 & \quad & \quad & \quad & \\[6mm]
1 & 15 & -77 & 13 & -36 \\
\quad & 4 & 76 & -4 & \\
\hline
1 & 19 & -1 & &
\end{array}
$$[/tex]
9. Add the fourth coefficient (13) and -4:
[tex]$$13 + (-4) = 9.$$[/tex]
Write 9 under the line.
[tex]$$
\begin{array}{cccccc}
\quad & 4 & \quad & \quad & \quad & \\[6mm]
1 & 15 & -77 & 13 & -36 \\
\quad & 4 & 76 & -4 & \\
\hline
1 & 19 & -1 & 9 &
\end{array}
$$[/tex]
10. Multiply 9 by [tex]$4$[/tex]:
[tex]$$9 \times 4 = 36.$$[/tex]
Write 36 under the fifth coefficient (-36).
[tex]$$
\begin{array}{cccccc}
\quad & 4 & \quad & \quad & \quad & \\[6mm]
1 & 15 & -77 & 13 & -36 \\
\quad & 4 & 76 & -4 & 36 \\
\hline
1 & 19 & -1 & 9 &
\end{array}
$$[/tex]
11. Finally, add the fifth coefficient (-36) and 36:
[tex]$$-36 + 36 = 0.$$[/tex]
Write 0 as the remainder.
[tex]$$
\begin{array}{cccccc}
\quad & 4 & \quad & \quad & \quad & \\[6mm]
1 & 15 & -77 & 13 & -36 \\
\quad & 4 & 76 & -4 & 36 \\
\hline
1 & 19 & -1 & 9 & 0
\end{array}
$$[/tex]
The numbers on the bottom row (except for the last number which is the remainder) are the coefficients of the quotient polynomial. Thus, the quotient is
[tex]$$
x^3 + 19x^2 - x + 9
$$[/tex]
and the remainder is [tex]$0$[/tex].
Therefore, the result is:
[tex]$$
\frac{x^4+15x^3-77x^2+13x-36}{x-4} = x^3+19x^2-x+9.
$$[/tex]
