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Sagot :
To determine which option correctly presents two ways to factor the expression [tex]\(18a - 36b + 90\)[/tex], we need to conduct a detailed examination of each option and compare them with the standard factored forms we know to be correct.
First, consider the original expression:
[tex]\[ 18a - 36b + 90 \][/tex]
### Factoring by taking common factors
We can factor out a common factor from each term in the expression.
Notice that each term in the expression has a common factor of [tex]\(18\)[/tex]:
[tex]\[ 18a - 36b + 90 = 18(a - 2b + 5) \][/tex]
So, one way to factor the expression is:
[tex]\[ 18(a - 2b + 5) \][/tex]
### Considering the negative factor
Now, we also need to provide another way to factor this expression, which includes factoring a negative common factor out of the expression:
[tex]\[ 18a - 36b + 90 = -18(-a + 2b - 5) \][/tex]
So, another way to factor the expression is:
[tex]\[ -18(-a + 2b - 5) \][/tex]
Given that these two forms are what we need to look for in the provided options:
### Checking the Options
Option A:
[tex]\[ \begin{array}{l} 9(-2a + 4b - 10) \\ -9(2a - 4b + 10) \end{array} \][/tex]
When simplified:
[tex]\[ 9(-2a + 4b - 10) = -18a + 36b - 90 \][/tex]
[tex]\[ -9(2a - 4b + 10) = -18a + 36b - 90 \][/tex]
These do not match the original expression [tex]\(18a - 36b + 90\)[/tex]. So, option A is incorrect.
Option B:
[tex]\[ \begin{array}{l} 6(3a - 6b + 15) \\ -6(-3a + 6b - 15) \end{array} \][/tex]
When simplified:
[tex]\[ 6(3a - 6b + 15) = 18a - 36b + 90 \][/tex]
[tex]\[ -6(-3a + 6b - 15) = 18a - 36b + 90 \][/tex]
These correctly simplify to the original expression [tex]\(18a - 36b + 90\)[/tex]. So, option B is correct.
Option C:
[tex]\[ \begin{array}{l} 9(2a - 4b + 10) \\ -9(-2a + 4b + 10) \end{array} \][/tex]
When simplified:
[tex]\[ 9(2a - 4b + 10) = 18a - 36b + 90 \][/tex]
[tex]\[ -9(-2a + 4b + 10) = 18a - 36b - 90 \][/tex]
The second part does not match the original expression [tex]\(18a - 36b + 90\)[/tex]. So, option C is incorrect.
Option D:
[tex]\[ \begin{array}{l} 6(-3a + 6b - 15) \\ -6(3a - 6b + 15) \end{array} \][/tex]
When simplified:
[tex]\[ 6(-3a + 6b - 15) = -18a + 36b - 90 \][/tex]
[tex]\[ -6(3a - 6b + 15) = -18a + 36b - 90 \][/tex]
These do not match the original expression [tex]\(18a - 36b + 90\)[/tex]. So, option D is incorrect.
### Conclusion:
The correct answer is B as it accurately shows two ways to factor the expression [tex]\(18a - 36b + 90\)[/tex]:
[tex]\[ \begin{array}{l} 6(3a - 6b + 15) \\ -6(-3a + 6b - 15) \end{array} \][/tex]
First, consider the original expression:
[tex]\[ 18a - 36b + 90 \][/tex]
### Factoring by taking common factors
We can factor out a common factor from each term in the expression.
Notice that each term in the expression has a common factor of [tex]\(18\)[/tex]:
[tex]\[ 18a - 36b + 90 = 18(a - 2b + 5) \][/tex]
So, one way to factor the expression is:
[tex]\[ 18(a - 2b + 5) \][/tex]
### Considering the negative factor
Now, we also need to provide another way to factor this expression, which includes factoring a negative common factor out of the expression:
[tex]\[ 18a - 36b + 90 = -18(-a + 2b - 5) \][/tex]
So, another way to factor the expression is:
[tex]\[ -18(-a + 2b - 5) \][/tex]
Given that these two forms are what we need to look for in the provided options:
### Checking the Options
Option A:
[tex]\[ \begin{array}{l} 9(-2a + 4b - 10) \\ -9(2a - 4b + 10) \end{array} \][/tex]
When simplified:
[tex]\[ 9(-2a + 4b - 10) = -18a + 36b - 90 \][/tex]
[tex]\[ -9(2a - 4b + 10) = -18a + 36b - 90 \][/tex]
These do not match the original expression [tex]\(18a - 36b + 90\)[/tex]. So, option A is incorrect.
Option B:
[tex]\[ \begin{array}{l} 6(3a - 6b + 15) \\ -6(-3a + 6b - 15) \end{array} \][/tex]
When simplified:
[tex]\[ 6(3a - 6b + 15) = 18a - 36b + 90 \][/tex]
[tex]\[ -6(-3a + 6b - 15) = 18a - 36b + 90 \][/tex]
These correctly simplify to the original expression [tex]\(18a - 36b + 90\)[/tex]. So, option B is correct.
Option C:
[tex]\[ \begin{array}{l} 9(2a - 4b + 10) \\ -9(-2a + 4b + 10) \end{array} \][/tex]
When simplified:
[tex]\[ 9(2a - 4b + 10) = 18a - 36b + 90 \][/tex]
[tex]\[ -9(-2a + 4b + 10) = 18a - 36b - 90 \][/tex]
The second part does not match the original expression [tex]\(18a - 36b + 90\)[/tex]. So, option C is incorrect.
Option D:
[tex]\[ \begin{array}{l} 6(-3a + 6b - 15) \\ -6(3a - 6b + 15) \end{array} \][/tex]
When simplified:
[tex]\[ 6(-3a + 6b - 15) = -18a + 36b - 90 \][/tex]
[tex]\[ -6(3a - 6b + 15) = -18a + 36b - 90 \][/tex]
These do not match the original expression [tex]\(18a - 36b + 90\)[/tex]. So, option D is incorrect.
### Conclusion:
The correct answer is B as it accurately shows two ways to factor the expression [tex]\(18a - 36b + 90\)[/tex]:
[tex]\[ \begin{array}{l} 6(3a - 6b + 15) \\ -6(-3a + 6b - 15) \end{array} \][/tex]
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