Sure, let's work through the problem step by step using synthetic division.
### Step 1: Set up the synthetic division
For synthetic division, we start by listing the coefficients of the polynomial. The polynomial given is [tex]\( 7x^3 + 0x^2 + 4x + 8 \)[/tex].
So, the coefficients are:
[tex]\[ 7, 0, 4, 8 \][/tex]
We are dividing by [tex]\( x + 2 \)[/tex]. In synthetic division, we use the opposite sign of the constant term from the divisor. So, we use [tex]\(-2\)[/tex] for the divisor.
### Step 2: Perform the synthetic division
1. Write down the coefficients:
[tex]\[ 7 \quad 0 \quad 4 \quad 8 \][/tex]
Write the divisor on the left:
[tex]\[
-2 \quad | \quad 7 \quad 0 \quad 4 \quad 8
\][/tex]
2. Bring down the first coefficient of the numerator (which is 7 in this case):
[tex]\[
-2 \quad | \quad 7 \quad 0 \quad 4 \quad 8 \\
\quad \quad \quad 7
\][/tex]
3. Multiply the divisor by this coefficient and write the result under the next coefficient:
[tex]\[
-2 \quad | \quad 7 \quad 0 \quad 4 \quad 8 \\
\quad \quad \quad 7 \quad (-2 \cdot 7) = -14
\][/tex]
4. Add the result to the next coefficient:
[tex]\[
-2 \quad | \quad 7 \quad 0 \quad 4 \quad 8 \\
\quad \quad \quad 7 \quad -14 \\
\quad \quad \quad 7 \quad -14 \quad (0-14) = -14
\][/tex]
5. Repeat the process of multiplying and adding:
[tex]\[
-2 \quad | \quad 7 \quad 0 \quad 4 \quad 8 \\
\quad \quad \quad 7 \quad -14 \quad (-2 \cdot -14) = 28
\][/tex]
[tex]\[
-2 \quad | \quad 7 \quad 0 \quad 4 \quad 8 \\
\quad \quad \quad 7 \quad -14 \quad 28 \\
\quad \quad \quad 7 \quad -14 \quad 28 \quad (4+28) = 32
\][/tex]
6. Finally, multiply and add one last time:
[tex]\[
-2 \quad | \quad 7 \quad 0 \quad 4 \quad 8 \\
\quad \quad \quad 7 \quad -14 \quad 28 \quad (-2 \cdot 32) = -64
\][/tex]
[tex]\[
-2 \quad | \quad 7 \quad 0 \quad 4 \quad 8 \\
\quad \quad \quad 7 \quad -14 \quad 28 \quad 32 \\
\quad \quad \quad 7 \quad -14 \quad 28 \quad 32 \quad (8-64) = -56
\][/tex]
### Step 3: Read the result
The numbers at the bottom (other than the last number) are the coefficients of the quotient polynomial. The last number is the remainder.
So, after synthetic division, the quotient polynomial is:
[tex]\[ 7x^2 - 14x + 32 \][/tex]
with a remainder of:
[tex]\[ -56 \][/tex]
Therefore, the result of the division [tex]\( \left(7 x^3 + 4 x + 8 \right) \div (x + 2) \)[/tex] is:
[tex]\[ 7x^2 - 14x + 32 \][/tex]
with a remainder:
[tex]\[ -56 \][/tex]
