Divide using synthetic division. Write your answer in standard form.

Question

(
5
x
4
+
12
x
3
−
21
x
2
−
40
x
−
12
)

÷

(
x
+
3
)
(5x
4
+12x
3
−21x
2
−40x−12) ÷ (x+3)

Answer 1

To perform synthetic division of the polynomial $5x^4+12x^3-21x^2-40x-12$ by $x+3$, we will use the root of $x+3$, which is $-3$.

1. Set up the synthetic division:

   • Write the coefficients of the polynomial: $5,12,-21,-40,-12$.
   • Use $-3$ as the divisor.

2. Synthetic division steps:

   • Write down the coefficients:

   $$\begin{array}{rrrrrrrrrrr}-3& 5& 12& -21& -40& -12& & & & & \end{array}$$

   • Bring down the leading coefficient (5):

   $$\begin{array}{rrrrrrrrrrrrrrrr}-3& 5& 12& -21& -40& -12& & & & & & 5& & & & \end{array}$$

   • Multiply $5$ by $-3$ and write the result under the next coefficient:

   $$\begin{array}{rrrrrrrrrrrrrrrr}-3& 5& 12& -21& -40& -12& & -15& & & & 5& & & & \end{array}$$

   • Add the second column:

   $$\begin{array}{rrrrrrrrrrrrrrrr}-3& 5& 12& -21& -40& -12& & -15& & & & 5& -3& & & \end{array}$$

   • Continue this process:

   $$-3\times -3=9\text{(add to -21:)}-21+9=-12$$

   • Continue this:

   $$\begin{array}{rrrrrrrrrrrrrrrr}-3& 5& 12& -21& -40& -12& & -15& 9& & & 5& -3& -12& & \end{array}$$

   • Multiply $-12$ by $-3$:

   $$-12\times -3=36$$

   • Add to -40:

   $$-40+36=-4$$

   $$\begin{array}{rrrrrrrrrrrrrrrr}-3& 5& 12& -21& -40& -12& & -15& 9& 36& & 5& -3& -12& -4& \end{array}$$

   • Finally, multiply $-4$ by $-3$:

   $$-4\times -3=12$$

   • Add to -12:

   $$-12+12=0$$

3. Complete synthetic division:

   The synthetic division is complete:

   $$\begin{array}{rrrrrrrrrrrrrrrr}-3& 5& 12& -21& -40& -12& & -15& 9& 36& 12& 5& -3& -12& -4& 0\end{array}$$

   • The result of the synthetic division is $5x^3-3x^2-12x-4$ (with a remainder of 0).

4. Write the answer in standard form:

   Thus, the solution to the division of $5x^4+12x^3-21x^2-40x-12$ by $x+3$ is:

   $$5x^3-3x^2-12x-4$$