To find the polynomial equivalent to the expression [tex]\(\frac{z^2 + 4z - 12}{z - 2}\)[/tex], we can use polynomial division. Let's break down the division step by step.
1. Divide the leading term of the numerator ([tex]\(z^2\)[/tex]) by the leading term of the divisor ([tex]\(z\)[/tex]). This gives us:
[tex]\[
\frac{z^2}{z} = z
\][/tex]
So, the first term in the quotient is [tex]\(z\)[/tex].
2. Multiply the entire divisor ([tex]\(z - 2\)[/tex]) by this leading term of the quotient ([tex]\(z\)[/tex]):
[tex]\[
z \cdot (z - 2) = z^2 - 2z
\][/tex]
3. Subtract this product from the original polynomial ([tex]\(z^2 + 4z - 12\)[/tex]):
[tex]\[
(z^2 + 4z - 12) - (z^2 - 2z) = 4z - 2z - 12 = 6z - 12
\][/tex]
4. Now, divide the leading term of the resulting polynomial ([tex]\(6z\)[/tex]) by the leading term of the divisor ([tex]\(z\)[/tex]). This gives us:
[tex]\[
\frac{6z}{z} = 6
\][/tex]
So, the next term in the quotient is [tex]\(6\)[/tex].
5. Multiply the entire divisor ([tex]\(z - 2\)[/tex]) by this new term of the quotient ([tex]\(6\)[/tex]):
[tex]\[
6 \cdot (z - 2) = 6z - 12
\][/tex]
6. Subtract this product from the previous remainder ([tex]\(6z - 12\)[/tex]):
[tex]\[
(6z - 12) - (6z - 12) = 0
\][/tex]
Since we end up with no remainder, we have completed the division. The quotient of the division is:
[tex]\[
z + 6
\][/tex]
Thus, the polynomial equivalent to [tex]\(\frac{z^2 + 4z - 12}{z - 2}\)[/tex] is:
[tex]\[
\boxed{z + 6}
\][/tex]
Therefore, the correct answer is:
B. [tex]\(z + 6\)[/tex]
