To divide \(x^3 - 6\) by \(x + 2\), we use polynomial division. Here is the step-by-step process:
1. Set up the division:
[tex]\[
\frac{x^3 + 0x^2 + 0x - 6}{x + 2}
\][/tex]
This expression means we are dividing \(x^3 + 0x^2 + 0x - 6\) by \(x + 2\).
2. First term:
To find the first term of the quotient, divide the leading term of the dividend (\(x^3\)) by the leading term of the divisor (\(x\)):
[tex]\[
\frac{x^3}{x} = x^2
\][/tex]
Multiply \(x^2\) by the entire divisor (\(x + 2\)):
[tex]\[
x^2 \cdot (x + 2) = x^3 + 2x^2
\][/tex]
Subtract this product from the original dividend:
[tex]\[
(x^3 + 0x^2 + 0x - 6) - (x^3 + 2x^2) = -2x^2 + 0x - 6
\][/tex]
3. Second term:
Now, divide the leading term of the new polynomial (\(-2x^2\)) by the leading term of the divisor (\(x\)):
[tex]\[
\frac{-2x^2}{x} = -2x
\][/tex]
Multiply \(-2x\) by \(x + 2\):
[tex]\[
-2x \cdot (x + 2) = -2x^2 - 4x
\][/tex]
Subtract this product from the new polynomial:
[tex]\[
(-2x^2 + 0x - 6) - (-2x^2 - 4x) = 4x - 6
\][/tex]
4. Third term:
Next, divide the leading term of the new polynomial (\(4x\)) by the leading term of the divisor (\(x\)):
[tex]\[
\frac{4x}{x} = 4
\][/tex]
Multiply \(4\) by \(x + 2\):
[tex]\[
4 \cdot (x + 2) = 4x + 8
\][/tex]
Subtract this product from the new polynomial:
[tex]\[
(4x - 6) - (4x + 8) = -14
\][/tex]
Thus, the quotient of the division is \(x^2 - 2x + 4\) and the remainder is \(-14\).
Putting everything together, we get:
[tex]\[
\frac{x^3 - 6}{x + 2} = x^2 - 2x + 4 + \frac{-14}{x + 2}
\][/tex]
Simplifying the expression gives us:
[tex]\[
x^2 - 2x + 4 - \frac{14}{x + 2}
\][/tex]
Therefore, the answer is:
[tex]\[
x^2 - 2x + 4 - \frac{14}{x + 2}
\][/tex]
