Answer :
Certainly! To divide the polynomial [tex]\(\frac{12x^{10} + 20x^9 + 4x^8}{4x^8}\)[/tex], we can carry out the division by handling each term in the numerator separately. Here's how you do it step by step:
1. Divide Each Term: We start by dividing each term in the numerator by the monomial in the denominator.
- The first term is [tex]\(\frac{12x^{10}}{4x^8}\)[/tex]. Divide the coefficients (12 by 4) and subtract the exponents (10 minus 8 for [tex]\(x\)[/tex]):
[tex]\[
\frac{12x^{10}}{4x^8} = 3x^2
\][/tex]
- The second term is [tex]\(\frac{20x^9}{4x^8}\)[/tex]. Similarly, divide the coefficients (20 by 4) and subtract the exponents (9 minus 8):
[tex]\[
\frac{20x^9}{4x^8} = 5x
\][/tex]
- The third term is [tex]\(\frac{4x^8}{4x^8}\)[/tex]. Since both the coefficients (4 divided by 4) and the exponents (8 minus 8) result in a power of zero for [tex]\(x\)[/tex], the result is:
[tex]\[
\frac{4x^8}{4x^8} = 1
\][/tex]
2. Combine the Results: Now, combine all the simplified terms:
[tex]\[
3x^2 + 5x + 1
\][/tex]
So, the simplified form of the expression [tex]\(\frac{12x^{10} + 20x^9 + 4x^8}{4x^8}\)[/tex] is [tex]\(3x^2 + 5x + 1\)[/tex].
1. Divide Each Term: We start by dividing each term in the numerator by the monomial in the denominator.
- The first term is [tex]\(\frac{12x^{10}}{4x^8}\)[/tex]. Divide the coefficients (12 by 4) and subtract the exponents (10 minus 8 for [tex]\(x\)[/tex]):
[tex]\[
\frac{12x^{10}}{4x^8} = 3x^2
\][/tex]
- The second term is [tex]\(\frac{20x^9}{4x^8}\)[/tex]. Similarly, divide the coefficients (20 by 4) and subtract the exponents (9 minus 8):
[tex]\[
\frac{20x^9}{4x^8} = 5x
\][/tex]
- The third term is [tex]\(\frac{4x^8}{4x^8}\)[/tex]. Since both the coefficients (4 divided by 4) and the exponents (8 minus 8) result in a power of zero for [tex]\(x\)[/tex], the result is:
[tex]\[
\frac{4x^8}{4x^8} = 1
\][/tex]
2. Combine the Results: Now, combine all the simplified terms:
[tex]\[
3x^2 + 5x + 1
\][/tex]
So, the simplified form of the expression [tex]\(\frac{12x^{10} + 20x^9 + 4x^8}{4x^8}\)[/tex] is [tex]\(3x^2 + 5x + 1\)[/tex].
