To divide the polynomial [tex]\( x^3 - 4 \)[/tex] by [tex]\( x + 2 \)[/tex], we will perform polynomial long division.
Here are the steps involved in performing the division:
1. Divide the leading term: Divide the leading term of the dividend (which is [tex]\( x^3 \)[/tex]) by the leading term of the divisor (which is [tex]\( x \)[/tex]). This gives [tex]\( \frac{x^3}{x} = x^2 \)[/tex].
2. Multiply and subtract: Multiply [tex]\( x^2 \)[/tex] by the divisor [tex]\( x + 2 \)[/tex], which gives [tex]\( x^3 + 2x^2 \)[/tex]. Subtract this product from the dividend:
[tex]\[
(x^3 - 4) - (x^3 + 2x^2) = -2x^2 - 4
\][/tex]
3. Repeat the process: Now, divide the new leading term [tex]\( -2x^2 \)[/tex] by the leading term of the divisor [tex]\( x \)[/tex]. This gives [tex]\( \frac{-2x^2}{x} = -2x \)[/tex].
4. Multiply and subtract: Multiply [tex]\( -2x \)[/tex] by the divisor [tex]\( x + 2 \)[/tex], which gives [tex]\( -2x^2 - 4x \)[/tex]. Subtract this product from the new dividend:
[tex]\[
(-2x^2 - 4) - (-2x^2 - 4x) = 4x - 4
\][/tex]
5. Repeat the process again: Now, divide the new leading term [tex]\( 4x \)[/tex] by the leading term of the divisor [tex]\( x \)[/tex]. This gives [tex]\( \frac{4x}{x} = 4 \)[/tex].
6. Multiply and subtract: Multiply [tex]\( 4 \)[/tex] by the divisor [tex]\( x + 2 \)[/tex], which gives [tex]\( 4x + 8 \)[/tex]. Subtract this product from [tex]\( 4x - 4 \)[/tex]:
[tex]\[
(4x - 4) - (4x + 8) = -12
\][/tex]
Therefore, the quotient is [tex]\( x^2 - 2x + 4 \)[/tex] and the remainder is [tex]\( -12 \)[/tex].
The result of the division is:
[tex]\[
\frac{x^3 - 4}{x + 2} = x^2 - 2x + 4 + \frac{-12}{x + 2}
\][/tex]
Which simplifies to:
[tex]\[
x^2 - 2x + 4 - \frac{12}{x + 2}
\][/tex]
Thus, the correct answer is:
[tex]\[
x^2-2x+4-\frac{12}{x+2}
\][/tex]
