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Answer :
Let's factorize each of the given expressions one by one and prove their solutions, identifying each case with its name.
### Expression 12: [tex]\( 16a^2 - 8ax + 20a^3x^2 - 12a^4x^3 \)[/tex]
To factorize the expression:
[tex]\[ 16a^2 - 8ax + 20a^3x^2 - 12a^4x^3 \][/tex]
We can take out the common factor:
[tex]\[ -4a(3a^3x^3 - 5a^2x^2 - 4a + 2x) \][/tex]
### Expression 13: [tex]\( (x + 1)(x + 2) + 9y(x + 2) \)[/tex]
To factorize the expression:
[tex]\[ (x + 1)(x + 2) + 9y(x + 2) \][/tex]
We can factor out the common term [tex]\( (x + 2) \)[/tex]:
[tex]\[ (x + 2)(x + 9y + 1) \][/tex]
### Expression 14: [tex]\( 4a^3x - 4a^2b + 33bm - 3amx \)[/tex]
To factorize the expression:
[tex]\[ 4a^3x - 4a^2b - 3amx + 33bm \][/tex]
We can group the terms:
[tex]\[ 4a^3 x - 3 am x - 4 a^2 b + 33 bm \][/tex]
Notice that it doesn't factor further nicely, so we leave it in this form.
### Expression 15: [tex]\( 400x^{10} + 40x^5 + 1 \)[/tex]
To factorize the expression:
[tex]\[ 400x^{10} + 40x^5 + 1 \][/tex]
Recognize it as a perfect square:
[tex]\[ (20x^5 + 1)^2 \][/tex]
### Expression 16: [tex]\( 100 - x^2 y^3 \)[/tex]
To factorize the expression:
[tex]\[ 100 - x^2 y^3 \][/tex]
Note that it can be written as:
[tex]\[ -x^2 y^3 + 100 \][/tex]
### Expression 17: [tex]\( x^2 - 17x + 60 \)[/tex]
To factorize the expression:
[tex]\[ x^2 - 17x + 60 \][/tex]
We look for two numbers that multiply to 60 and add to -17:
[tex]\[ (x - 12)(x - 5) \][/tex]
### Expression 18: [tex]\( 20x^2 - 9x - 20 \)[/tex]
To factorize the expression:
[tex]\[ 20x^2 - 9x - 20 \][/tex]
We look for two numbers that multiply to [tex]\( 20 \times -20 = -400 \)[/tex] and add to -9:
[tex]\[ (4x - 5)(5x + 4) \][/tex]
### Expression 19: [tex]\( 216 - 756a^2 + 882a^4 - 343a^6 \)[/tex]
To factorize the expression:
[tex]\[ 216 - 756a^2 + 882a^4 - 343a^6 \][/tex]
Recognize it as a difference of cubes:
[tex]\[ -(7a^2 - 6)^3 \][/tex]
### Expression 20: [tex]\( 8a^3 + 27b^6 \)[/tex]
To factorize the expression:
[tex]\[ 8a^3 + 27b^6 \][/tex]
Recognize it as the sum of cubes:
[tex]\[ (2a + 3b^2)(4a^2 - 6ab^2 + 9b^4) \][/tex]
By following these steps, we have factorized each expression and identified the appropriate cases for each.
### Expression 12: [tex]\( 16a^2 - 8ax + 20a^3x^2 - 12a^4x^3 \)[/tex]
To factorize the expression:
[tex]\[ 16a^2 - 8ax + 20a^3x^2 - 12a^4x^3 \][/tex]
We can take out the common factor:
[tex]\[ -4a(3a^3x^3 - 5a^2x^2 - 4a + 2x) \][/tex]
### Expression 13: [tex]\( (x + 1)(x + 2) + 9y(x + 2) \)[/tex]
To factorize the expression:
[tex]\[ (x + 1)(x + 2) + 9y(x + 2) \][/tex]
We can factor out the common term [tex]\( (x + 2) \)[/tex]:
[tex]\[ (x + 2)(x + 9y + 1) \][/tex]
### Expression 14: [tex]\( 4a^3x - 4a^2b + 33bm - 3amx \)[/tex]
To factorize the expression:
[tex]\[ 4a^3x - 4a^2b - 3amx + 33bm \][/tex]
We can group the terms:
[tex]\[ 4a^3 x - 3 am x - 4 a^2 b + 33 bm \][/tex]
Notice that it doesn't factor further nicely, so we leave it in this form.
### Expression 15: [tex]\( 400x^{10} + 40x^5 + 1 \)[/tex]
To factorize the expression:
[tex]\[ 400x^{10} + 40x^5 + 1 \][/tex]
Recognize it as a perfect square:
[tex]\[ (20x^5 + 1)^2 \][/tex]
### Expression 16: [tex]\( 100 - x^2 y^3 \)[/tex]
To factorize the expression:
[tex]\[ 100 - x^2 y^3 \][/tex]
Note that it can be written as:
[tex]\[ -x^2 y^3 + 100 \][/tex]
### Expression 17: [tex]\( x^2 - 17x + 60 \)[/tex]
To factorize the expression:
[tex]\[ x^2 - 17x + 60 \][/tex]
We look for two numbers that multiply to 60 and add to -17:
[tex]\[ (x - 12)(x - 5) \][/tex]
### Expression 18: [tex]\( 20x^2 - 9x - 20 \)[/tex]
To factorize the expression:
[tex]\[ 20x^2 - 9x - 20 \][/tex]
We look for two numbers that multiply to [tex]\( 20 \times -20 = -400 \)[/tex] and add to -9:
[tex]\[ (4x - 5)(5x + 4) \][/tex]
### Expression 19: [tex]\( 216 - 756a^2 + 882a^4 - 343a^6 \)[/tex]
To factorize the expression:
[tex]\[ 216 - 756a^2 + 882a^4 - 343a^6 \][/tex]
Recognize it as a difference of cubes:
[tex]\[ -(7a^2 - 6)^3 \][/tex]
### Expression 20: [tex]\( 8a^3 + 27b^6 \)[/tex]
To factorize the expression:
[tex]\[ 8a^3 + 27b^6 \][/tex]
Recognize it as the sum of cubes:
[tex]\[ (2a + 3b^2)(4a^2 - 6ab^2 + 9b^4) \][/tex]
By following these steps, we have factorized each expression and identified the appropriate cases for each.
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