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Sagot :
Let's determine the Greatest Common Monomial Factor (GCMF) of each polynomial and write its factored form step-by-step.
### 1. [tex]\( x^2 + 2x \)[/tex]
- GCMF:
- The common factor between [tex]\( x^2 \)[/tex] and [tex]\( 2x \)[/tex] is [tex]\( x \)[/tex].
- Remaining Factor:
- [tex]\( \frac{x^2}{x} = x \)[/tex]
- [tex]\( \frac{2x}{x} = 2 \)[/tex]
- Factored Form:
- [tex]\( x^2 + 2x = x(x + 2) \)[/tex]
### 2. [tex]\( 5x^2 - 10x^3 \)[/tex]
- GCMF:
- The common factor between [tex]\( 5x^2 \)[/tex] and [tex]\( 10x^3 \)[/tex] is [tex]\( 5x^2 \)[/tex].
- Remaining Factor:
- [tex]\( \frac{5x^2}{5x^2} = 1 \)[/tex]
- [tex]\( \frac{-10x^3}{5x^2} = -2x \)[/tex]
- Factored Form:
- [tex]\( 5x^2 - 10x^3 = 5x^2(1 - 2x) \)[/tex]
### 3. [tex]\( 25x^2y^3 + 55xy^3 \)[/tex]
- GCMF:
- The common factor between [tex]\( 25x^2y^3 \)[/tex] and [tex]\( 55xy^3 \)[/tex] is [tex]\( 5xy^3 \)[/tex].
- Remaining Factor:
- [tex]\( \frac{25x^2y^3}{5xy^3} = 5x \)[/tex]
- [tex]\( \frac{55xy^3}{5xy^3} = 11 \)[/tex]
- Factored Form:
- [tex]\( 25x^2y^3 + 55xy^3 = 5xy^3(5x + 11) \)[/tex]
### 4. [tex]\( 10c^3 - 80c^5 - 5c^6 + 5c^7 \)[/tex]
- GCMF:
- The common factor among [tex]\( 10c^3, 80c^5, -5c^6, 5c^7 \)[/tex] is [tex]\( 5c^3 \)[/tex].
- Remaining Factor:
- [tex]\( \frac{10c^3}{5c^3} = 2 \)[/tex]
- [tex]\( \frac{80c^5}{5c^3} = 16c^2 \)[/tex]
- [tex]\( \frac{-5c^6}{5c^3} = -c^3 \)[/tex]
- [tex]\( \frac{5c^7}{5c^3} = c^4 \)[/tex]
- Factored Form:
- [tex]\( 10c^3 - 80c^5 - 5c^6 + 5c^7 = 5c^3(2 - 16c^2 - c^3 + c^4) \)[/tex]
### 5. [tex]\( 12m^5n^2 - 6m^2n^3 + 3mn \)[/tex]
- GCMF:
- The common factor among [tex]\( 12m^5n^2, 6m^2n^3, 3mn \)[/tex] is [tex]\( 3mn \)[/tex].
- Remaining Factor:
- [tex]\( \frac{12m^5n^2}{3mn} = 4m^4n \)[/tex]
- [tex]\( \frac{6m^2n^3}{3mn} = 2m \)[/tex]
- [tex]\( \frac{3mn}{3mn} = 1 \)[/tex]
- Factored Form:
- [tex]\( 12m^5n^2 - 6m^2n^3 + 3mn = 3mn(4m^4n - 2m + 1) \)[/tex]
Now, let's summarize the factored forms in a table:
[tex]\[ \begin{array}{|c|c|c|c|} \hline \text{Polynomial} & \text{GCMF} & \text{Remaining Factor} & \text{Factored Form} \\ \hline x^2 + 2x & x & x + 2 & x(x + 2) \\ \hline 5x^2 - 10x^3 & 5x^2 & 1 - 2x & 5x^2(1 - 2x) \\ \hline 25x^2y^3 + 55xy^3 & 5xy^3 & 5x + 11 & 5xy^3(5x + 11) \\ \hline 10c^3 - 80c^5 - 5c^6 + 5c^7 & 5c^3 & 2 - 16c^2 - c^3 + c^4 & 5c^3(2 - 16c^2 - c^3 + c^4) \\ \hline 12m^5n^2 - 6m^2n^3 + 3mn & 3mn & 4m^4n - 2m + 1 & 3mn(4m^4n - 2m + 1) \\ \hline \end{array} \][/tex]
### 1. [tex]\( x^2 + 2x \)[/tex]
- GCMF:
- The common factor between [tex]\( x^2 \)[/tex] and [tex]\( 2x \)[/tex] is [tex]\( x \)[/tex].
- Remaining Factor:
- [tex]\( \frac{x^2}{x} = x \)[/tex]
- [tex]\( \frac{2x}{x} = 2 \)[/tex]
- Factored Form:
- [tex]\( x^2 + 2x = x(x + 2) \)[/tex]
### 2. [tex]\( 5x^2 - 10x^3 \)[/tex]
- GCMF:
- The common factor between [tex]\( 5x^2 \)[/tex] and [tex]\( 10x^3 \)[/tex] is [tex]\( 5x^2 \)[/tex].
- Remaining Factor:
- [tex]\( \frac{5x^2}{5x^2} = 1 \)[/tex]
- [tex]\( \frac{-10x^3}{5x^2} = -2x \)[/tex]
- Factored Form:
- [tex]\( 5x^2 - 10x^3 = 5x^2(1 - 2x) \)[/tex]
### 3. [tex]\( 25x^2y^3 + 55xy^3 \)[/tex]
- GCMF:
- The common factor between [tex]\( 25x^2y^3 \)[/tex] and [tex]\( 55xy^3 \)[/tex] is [tex]\( 5xy^3 \)[/tex].
- Remaining Factor:
- [tex]\( \frac{25x^2y^3}{5xy^3} = 5x \)[/tex]
- [tex]\( \frac{55xy^3}{5xy^3} = 11 \)[/tex]
- Factored Form:
- [tex]\( 25x^2y^3 + 55xy^3 = 5xy^3(5x + 11) \)[/tex]
### 4. [tex]\( 10c^3 - 80c^5 - 5c^6 + 5c^7 \)[/tex]
- GCMF:
- The common factor among [tex]\( 10c^3, 80c^5, -5c^6, 5c^7 \)[/tex] is [tex]\( 5c^3 \)[/tex].
- Remaining Factor:
- [tex]\( \frac{10c^3}{5c^3} = 2 \)[/tex]
- [tex]\( \frac{80c^5}{5c^3} = 16c^2 \)[/tex]
- [tex]\( \frac{-5c^6}{5c^3} = -c^3 \)[/tex]
- [tex]\( \frac{5c^7}{5c^3} = c^4 \)[/tex]
- Factored Form:
- [tex]\( 10c^3 - 80c^5 - 5c^6 + 5c^7 = 5c^3(2 - 16c^2 - c^3 + c^4) \)[/tex]
### 5. [tex]\( 12m^5n^2 - 6m^2n^3 + 3mn \)[/tex]
- GCMF:
- The common factor among [tex]\( 12m^5n^2, 6m^2n^3, 3mn \)[/tex] is [tex]\( 3mn \)[/tex].
- Remaining Factor:
- [tex]\( \frac{12m^5n^2}{3mn} = 4m^4n \)[/tex]
- [tex]\( \frac{6m^2n^3}{3mn} = 2m \)[/tex]
- [tex]\( \frac{3mn}{3mn} = 1 \)[/tex]
- Factored Form:
- [tex]\( 12m^5n^2 - 6m^2n^3 + 3mn = 3mn(4m^4n - 2m + 1) \)[/tex]
Now, let's summarize the factored forms in a table:
[tex]\[ \begin{array}{|c|c|c|c|} \hline \text{Polynomial} & \text{GCMF} & \text{Remaining Factor} & \text{Factored Form} \\ \hline x^2 + 2x & x & x + 2 & x(x + 2) \\ \hline 5x^2 - 10x^3 & 5x^2 & 1 - 2x & 5x^2(1 - 2x) \\ \hline 25x^2y^3 + 55xy^3 & 5xy^3 & 5x + 11 & 5xy^3(5x + 11) \\ \hline 10c^3 - 80c^5 - 5c^6 + 5c^7 & 5c^3 & 2 - 16c^2 - c^3 + c^4 & 5c^3(2 - 16c^2 - c^3 + c^4) \\ \hline 12m^5n^2 - 6m^2n^3 + 3mn & 3mn & 4m^4n - 2m + 1 & 3mn(4m^4n - 2m + 1) \\ \hline \end{array} \][/tex]
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