Certainly! To divide the polynomial [tex]\(x^2 + 2x + 6\)[/tex] by [tex]\(x - 3\)[/tex], we will use polynomial long division. Here's a step-by-step breakdown of how the division process works:
### Step 1: Set up the Division
We start by setting up the division, where [tex]\(x^2 + 2x + 6\)[/tex] (the dividend) is divided by [tex]\(x - 3\)[/tex] (the divisor).
```
_______________
x - 3 | x^2 + 2x + 6
```
### Step 2: Divide the Leading Terms
Divide the leading term of the dividend (which is [tex]\(x^2\)[/tex]) by the leading term of the divisor (which is [tex]\(x\)[/tex]):
[tex]\[
\frac{x^2}{x} = x
\][/tex]
So, [tex]\(x\)[/tex] is the first term of our quotient.
### Step 3: Multiply and Subtract
Now, multiply the entire divisor [tex]\(x - 3\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
x \cdot (x - 3) = x^2 - 3x
\][/tex]
Next, subtract this result from the original dividend:
[tex]\[
(x^2 + 2x + 6) - (x^2 - 3x) = (x^2 + 2x + 6) - x^2 + 3x = 5x + 6
\][/tex]
This leaves us with a new dividend:
[tex]\[
5x + 6
\][/tex]
### Step 4: Repeat the Process
Now, we repeat the process with the new dividend [tex]\(5x + 6\)[/tex].
First, divide the leading term of the new dividend (which is [tex]\(5x\)[/tex]) by the leading term of the divisor (which is [tex]\(x\)[/tex]):
[tex]\[
\frac{5x}{x} = 5
\][/tex]
So, [tex]\(5\)[/tex] is the next term of our quotient.
### Step 5: Multiply and Subtract Again
Multiply the entire divisor [tex]\(x - 3\)[/tex] by [tex]\(5\)[/tex]:
[tex]\[
5 \cdot (x - 3) = 5x - 15
\][/tex]
Subtract this result from the new dividend:
[tex]\[
(5x + 6) - (5x - 15) = (5x + 6) - 5x + 15 = 21
\][/tex]
This leaves us with:
[tex]\[
21
\][/tex]
### Step 6: Collect the Results
Since the degree of the remainder [tex]\(21\)[/tex] is less than the degree of the divisor [tex]\(x - 3\)[/tex], we cannot continue the division further.
Thus, the quotient is [tex]\(x + 5\)[/tex] and the remainder is [tex]\(21\)[/tex].
### Final Result
[tex]\[
\frac{x^2 + 2x + 6}{x - 3} = x + 5 + \frac{21}{x - 3}
\][/tex]
So, the quotient is [tex]\(x + 5\)[/tex] and the remainder is [tex]\(21\)[/tex].
