Answer :
Let's find the completely factored form of the expression [tex]\( 9x^3y - 100xy \)[/tex].
### Step-by-Step Solution
1. Identify Common Factors: Look for common factors in the terms [tex]\( 9x^3y \)[/tex] and [tex]\( 100xy \)[/tex].
Both terms have [tex]\( xy \)[/tex] as a common factor.
2. Factor Out the Common Factor:
Factor [tex]\( xy \)[/tex] out from both terms:
[tex]\[ 9x^3y - 100xy = xy(9x^2 - 100) \][/tex]
3. Factor the Quadratic Expression: Now factor the quadratic expression [tex]\( 9x^2 - 100 \)[/tex].
Notice that [tex]\( 9x^2 - 100 \)[/tex] is a difference of squares. Recall that a difference of squares [tex]\( a^2 - b^2 \)[/tex] can be factored as [tex]\( (a - b)(a + b) \)[/tex].
Here, [tex]\( 9x^2 \)[/tex] is a perfect square ([tex]\( (3x)^2 \)[/tex]), and [tex]\( 100 \)[/tex] is also a perfect square ([tex]\( 10^2 \)[/tex]). So we can write:
[tex]\[ 9x^2 - 100 = (3x)^2 - 10^2 \][/tex]
4. Apply the Difference of Squares Formula:
Using the formula [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex]:
[tex]\[ (3x)^2 - 10^2 = (3x - 10)(3x + 10) \][/tex]
5. Combine Everything: Substitute back into the factored expression:
[tex]\[ 9x^3y - 100xy = xy(3x - 10)(3x + 10) \][/tex]
Thus, the completely factored form of the given expression [tex]\( 9x^3y - 100xy \)[/tex] is:
[tex]\[ xy(3x - 10)(3x + 10) \][/tex]
### Correct Answer
[tex]\[ \boxed{xy(3x - 10)(3x + 10)} \][/tex]
So, the correct choice is:
[tex]\[ \boxed{x y (3 x - 10)(3 x + 10)} \][/tex]
### Step-by-Step Solution
1. Identify Common Factors: Look for common factors in the terms [tex]\( 9x^3y \)[/tex] and [tex]\( 100xy \)[/tex].
Both terms have [tex]\( xy \)[/tex] as a common factor.
2. Factor Out the Common Factor:
Factor [tex]\( xy \)[/tex] out from both terms:
[tex]\[ 9x^3y - 100xy = xy(9x^2 - 100) \][/tex]
3. Factor the Quadratic Expression: Now factor the quadratic expression [tex]\( 9x^2 - 100 \)[/tex].
Notice that [tex]\( 9x^2 - 100 \)[/tex] is a difference of squares. Recall that a difference of squares [tex]\( a^2 - b^2 \)[/tex] can be factored as [tex]\( (a - b)(a + b) \)[/tex].
Here, [tex]\( 9x^2 \)[/tex] is a perfect square ([tex]\( (3x)^2 \)[/tex]), and [tex]\( 100 \)[/tex] is also a perfect square ([tex]\( 10^2 \)[/tex]). So we can write:
[tex]\[ 9x^2 - 100 = (3x)^2 - 10^2 \][/tex]
4. Apply the Difference of Squares Formula:
Using the formula [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex]:
[tex]\[ (3x)^2 - 10^2 = (3x - 10)(3x + 10) \][/tex]
5. Combine Everything: Substitute back into the factored expression:
[tex]\[ 9x^3y - 100xy = xy(3x - 10)(3x + 10) \][/tex]
Thus, the completely factored form of the given expression [tex]\( 9x^3y - 100xy \)[/tex] is:
[tex]\[ xy(3x - 10)(3x + 10) \][/tex]
### Correct Answer
[tex]\[ \boxed{xy(3x - 10)(3x + 10)} \][/tex]
So, the correct choice is:
[tex]\[ \boxed{x y (3 x - 10)(3 x + 10)} \][/tex]
