Of course! Let's complete the table by finding the Common Monomial Factor (CMF) and the quotient of each polynomial when divided by the CMF.
### Polynomial: [tex]\( 5p^2 + 25p^4 \)[/tex]
1. Find the CMF:
- The common numerical factor between [tex]\( 5 \)[/tex] and [tex]\( 25 \)[/tex] is [tex]\( 5 \)[/tex].
- The common variable factor between [tex]\( p^2 \)[/tex] and [tex]\( p^4 \)[/tex] is [tex]\( p^2 \)[/tex].
- Therefore, the CMF is [tex]\( 5p^2 \)[/tex].
2. Find the Quotient:
- Divide each term of the polynomial by the CMF [tex]\( 5p^2 \)[/tex]:
[tex]\[
\frac{5p^2}{5p^2} = 1
\][/tex]
[tex]\[
\frac{25p^4}{5p^2} = 5p^2
\][/tex]
- Therefore, the quotient is [tex]\( 1 + 5p^2 \)[/tex].
### Polynomial: [tex]\( 3m - 12 \)[/tex]
1. Find the CMF:
- The common numerical factor between [tex]\( 3 \)[/tex] and [tex]\( -12 \)[/tex] is [tex]\( 3 \)[/tex].
- There is no common variable factor, so the CMF is just [tex]\( 3 \)[/tex].
2. Find the Quotient:
- Divide each term of the polynomial by the CMF [tex]\( 3 \)[/tex]:
[tex]\[
\frac{3m}{3} = m
\][/tex]
[tex]\[
\frac{-12}{3} = -4
\][/tex]
- Therefore, the quotient is [tex]\( m - 4 \)[/tex].
### Polynomial: [tex]\( 3x^2 - 12xy \)[/tex]
1. Find the CMF:
- The common numerical factor between [tex]\( 3 \)[/tex] and [tex]\( -12 \)[/tex] is [tex]\( 3 \)[/tex].
- The common variable factor between [tex]\( x^2 \)[/tex] and [tex]\( xy \)[/tex] is [tex]\( x \)[/tex].
- Therefore, the CMF is [tex]\( 3x \)[/tex].
2. Find the Quotient:
- Divide each term of the polynomial by the CMF [tex]\( 3x \)[/tex]:
[tex]\[
\frac{3x^2}{3x} = x
\][/tex]
[tex]\[
\frac{-12xy}{3x} = -4y
\][/tex]
- Therefore, the quotient is [tex]\( x - 4y \)[/tex].
### Completed Table
[tex]\[
\begin{tabular}{|l|l|l|l|}
\hline
Polynomial & CMF & \begin{tabular}{c}
Quotient of \\
Polynomial and CMF
\end{tabular} & \\
\hline
\( 5p^2 + 25p^4 \) & \( 5p^2 \) & \( 1 + 5p^2 \) & \\
\hline
\( 3m - 12 \) & \( 3 \) & \( m - 4 \) & \\
\hline
\( 3x^2 - 12xy \) & \( 3x \) & \( x - 4y \) & \\
\hline
\end{tabular}
\][/tex]
By identifying the CMF and performing the division for each polynomial, we obtained the quotients and completed the table.
