Answer :
To simplify the given expression [tex]\(\frac{4x^8 + 8x^6 - 12x^2}{4x^2}\)[/tex], we can perform the division separately for each term in the numerator. Here’s how you can simplify it step-by-step:
1. Divide each term in the numerator by the denominator:
- The first term is [tex]\(4x^8\)[/tex]. Divide [tex]\(4x^8\)[/tex] by [tex]\(4x^2\)[/tex]:
[tex]\[
\frac{4x^8}{4x^2} = x^{8-2} = x^6
\][/tex]
- The second term is [tex]\(8x^6\)[/tex]. Divide [tex]\(8x^6\)[/tex] by [tex]\(4x^2\)[/tex]:
[tex]\[
\frac{8x^6}{4x^2} = 2x^{6-2} = 2x^4
\][/tex]
- The third term is [tex]\(-12x^2\)[/tex]. Divide [tex]\(-12x^2\)[/tex] by [tex]\(4x^2\)[/tex]:
[tex]\[
\frac{-12x^2}{4x^2} = -3x^{2-2} = -3
\][/tex]
2. Combine the simplified terms:
After dividing each term by [tex]\(4x^2\)[/tex], we combine them to get the simplified expression:
[tex]\[
x^6 + 2x^4 - 3
\][/tex]
Therefore, the simplified expression is [tex]\(x^6 + 2x^4 - 3\)[/tex], which corresponds to option b).
1. Divide each term in the numerator by the denominator:
- The first term is [tex]\(4x^8\)[/tex]. Divide [tex]\(4x^8\)[/tex] by [tex]\(4x^2\)[/tex]:
[tex]\[
\frac{4x^8}{4x^2} = x^{8-2} = x^6
\][/tex]
- The second term is [tex]\(8x^6\)[/tex]. Divide [tex]\(8x^6\)[/tex] by [tex]\(4x^2\)[/tex]:
[tex]\[
\frac{8x^6}{4x^2} = 2x^{6-2} = 2x^4
\][/tex]
- The third term is [tex]\(-12x^2\)[/tex]. Divide [tex]\(-12x^2\)[/tex] by [tex]\(4x^2\)[/tex]:
[tex]\[
\frac{-12x^2}{4x^2} = -3x^{2-2} = -3
\][/tex]
2. Combine the simplified terms:
After dividing each term by [tex]\(4x^2\)[/tex], we combine them to get the simplified expression:
[tex]\[
x^6 + 2x^4 - 3
\][/tex]
Therefore, the simplified expression is [tex]\(x^6 + 2x^4 - 3\)[/tex], which corresponds to option b).
