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Sagot :
Certainly! To find two monomials whose product is a given monomial, we need to consider the different ways we can split the given monomial into two factors with positive integer coefficients. Let's go through each case step-by-step:
### (i) [tex]\( abc \)[/tex]
To factorize [tex]\( abc \)[/tex], we look for pairs of monomials whose product is [tex]\( abc \)[/tex]. The possible pairs are:
1. [tex]\( a \times bc \)[/tex]:
- First monomial: [tex]\( a \)[/tex]
- Second monomial: [tex]\( bc \)[/tex]
2. [tex]\( b \times ac \)[/tex]:
- First monomial: [tex]\( b \)[/tex]
- Second monomial: [tex]\( ac \)[/tex]
3. [tex]\( c \times ab \)[/tex]:
- First monomial: [tex]\( c \)[/tex]
- Second monomial: [tex]\( ab \)[/tex]
4. [tex]\( 1 \times abc \)[/tex]:
- First monomial: [tex]\( 1 \)[/tex]
- Second monomial: [tex]\( abc \)[/tex]
So, the factor pairs for [tex]\( abc \)[/tex] are:
[tex]\[ (a, bc), (b, ac), (c, ab), (1, abc) \][/tex]
### (ii) [tex]\( xyz \)[/tex]
To factorize [tex]\( xyz \)[/tex], we look for pairs of monomials whose product is [tex]\( xyz \)[/tex]. The possible pairs are:
1. [tex]\( x \times yz \)[/tex]:
- First monomial: [tex]\( x \)[/tex]
- Second monomial: [tex]\( yz \)[/tex]
2. [tex]\( y \times xz \)[/tex]:
- First monomial: [tex]\( y \)[/tex]
- Second monomial: [tex]\( xz \)[/tex]
3. [tex]\( z \times xy \)[/tex]:
- First monomial: [tex]\( z \)[/tex]
- Second monomial: [tex]\( xy \)[/tex]
4. [tex]\( 1 \times xyz \)[/tex]:
- First monomial: [tex]\( 1 \)[/tex]
- Second monomial: [tex]\( xyz \)[/tex]
So, the factor pairs for [tex]\( xyz \)[/tex] are:
[tex]\[ (x, yz), (y, xz), (z, xy), (1, xyz) \][/tex]
### (iii) [tex]\( a^2 b c \)[/tex]
To factorize [tex]\( a^2 b c \)[/tex], we look for pairs of monomials whose product is [tex]\( a^2 b c \)[/tex]. The possible pairs are:
1. [tex]\( a \times a bc \)[/tex]:
- First monomial: [tex]\( a \)[/tex]
- Second monomial: [tex]\( a bc \)[/tex]
2. [tex]\( a^2 \times bc \)[/tex]:
- First monomial: [tex]\( a^2 \)[/tex]
- Second monomial: [tex]\( bc \)[/tex]
3. [tex]\( b \times a^2 c \)[/tex]:
- First monomial: [tex]\( b \)[/tex]
- Second monomial: [tex]\( a^2 c \)[/tex]
4. [tex]\( c \times a^2 b \)[/tex]:
- First monomial: [tex]\( c \)[/tex]
- Second monomial: [tex]\( a^2 b \)[/tex]
5. [tex]\( 1 \times a^2 b c \)[/tex]:
- First monomial: [tex]\( 1 \)[/tex]
- Second monomial: [tex]\( a^2 b c \)[/tex]
So, the factor pairs for [tex]\( a^2 b c \)[/tex] are:
[tex]\[ (a, a bc), (a^2, bc), (b, a^2 c), (c, a^2 b), (1, a^2 b c) \][/tex]
### (iv) [tex]\( 7 a b c \)[/tex]
To factorize [tex]\( 7 a b c \)[/tex], we look for pairs of monomials whose product is [tex]\( 7 a b c \)[/tex]. The possible pairs are:
1. [tex]\( 7 \times a bc \)[/tex]:
- First monomial: [tex]\( 7 \)[/tex]
- Second monomial: [tex]\( a bc \)[/tex]
2. [tex]\( 7a \times bc \)[/tex]:
- First monomial: [tex]\( 7a \)[/tex]
- Second monomial: [tex]\( bc \)[/tex]
3. [tex]\( 7b \times ac \)[/tex]:
- First monomial: [tex]\( 7b \)[/tex]
- Second monomial: [tex]\( ac \)[/tex]
4. [tex]\( 7c \times ab \)[/tex]:
- First monomial: [tex]\( 7c \)[/tex]
- Second monomial: [tex]\( ab \)[/tex]
5. [tex]\( 1 \times 7 a b c \)[/tex]:
- First monomial: [tex]\( 1 \)[/tex]
- Second monomial: [tex]\( 7 a b c \)[/tex]
So, the factor pairs for [tex]\( 7 a b c \)[/tex] are:
[tex]\[ (7, a bc), (7a, bc), (7b, ac), (7c, ab), (1, 7 a b c) \][/tex]
### (i) [tex]\( abc \)[/tex]
To factorize [tex]\( abc \)[/tex], we look for pairs of monomials whose product is [tex]\( abc \)[/tex]. The possible pairs are:
1. [tex]\( a \times bc \)[/tex]:
- First monomial: [tex]\( a \)[/tex]
- Second monomial: [tex]\( bc \)[/tex]
2. [tex]\( b \times ac \)[/tex]:
- First monomial: [tex]\( b \)[/tex]
- Second monomial: [tex]\( ac \)[/tex]
3. [tex]\( c \times ab \)[/tex]:
- First monomial: [tex]\( c \)[/tex]
- Second monomial: [tex]\( ab \)[/tex]
4. [tex]\( 1 \times abc \)[/tex]:
- First monomial: [tex]\( 1 \)[/tex]
- Second monomial: [tex]\( abc \)[/tex]
So, the factor pairs for [tex]\( abc \)[/tex] are:
[tex]\[ (a, bc), (b, ac), (c, ab), (1, abc) \][/tex]
### (ii) [tex]\( xyz \)[/tex]
To factorize [tex]\( xyz \)[/tex], we look for pairs of monomials whose product is [tex]\( xyz \)[/tex]. The possible pairs are:
1. [tex]\( x \times yz \)[/tex]:
- First monomial: [tex]\( x \)[/tex]
- Second monomial: [tex]\( yz \)[/tex]
2. [tex]\( y \times xz \)[/tex]:
- First monomial: [tex]\( y \)[/tex]
- Second monomial: [tex]\( xz \)[/tex]
3. [tex]\( z \times xy \)[/tex]:
- First monomial: [tex]\( z \)[/tex]
- Second monomial: [tex]\( xy \)[/tex]
4. [tex]\( 1 \times xyz \)[/tex]:
- First monomial: [tex]\( 1 \)[/tex]
- Second monomial: [tex]\( xyz \)[/tex]
So, the factor pairs for [tex]\( xyz \)[/tex] are:
[tex]\[ (x, yz), (y, xz), (z, xy), (1, xyz) \][/tex]
### (iii) [tex]\( a^2 b c \)[/tex]
To factorize [tex]\( a^2 b c \)[/tex], we look for pairs of monomials whose product is [tex]\( a^2 b c \)[/tex]. The possible pairs are:
1. [tex]\( a \times a bc \)[/tex]:
- First monomial: [tex]\( a \)[/tex]
- Second monomial: [tex]\( a bc \)[/tex]
2. [tex]\( a^2 \times bc \)[/tex]:
- First monomial: [tex]\( a^2 \)[/tex]
- Second monomial: [tex]\( bc \)[/tex]
3. [tex]\( b \times a^2 c \)[/tex]:
- First monomial: [tex]\( b \)[/tex]
- Second monomial: [tex]\( a^2 c \)[/tex]
4. [tex]\( c \times a^2 b \)[/tex]:
- First monomial: [tex]\( c \)[/tex]
- Second monomial: [tex]\( a^2 b \)[/tex]
5. [tex]\( 1 \times a^2 b c \)[/tex]:
- First monomial: [tex]\( 1 \)[/tex]
- Second monomial: [tex]\( a^2 b c \)[/tex]
So, the factor pairs for [tex]\( a^2 b c \)[/tex] are:
[tex]\[ (a, a bc), (a^2, bc), (b, a^2 c), (c, a^2 b), (1, a^2 b c) \][/tex]
### (iv) [tex]\( 7 a b c \)[/tex]
To factorize [tex]\( 7 a b c \)[/tex], we look for pairs of monomials whose product is [tex]\( 7 a b c \)[/tex]. The possible pairs are:
1. [tex]\( 7 \times a bc \)[/tex]:
- First monomial: [tex]\( 7 \)[/tex]
- Second monomial: [tex]\( a bc \)[/tex]
2. [tex]\( 7a \times bc \)[/tex]:
- First monomial: [tex]\( 7a \)[/tex]
- Second monomial: [tex]\( bc \)[/tex]
3. [tex]\( 7b \times ac \)[/tex]:
- First monomial: [tex]\( 7b \)[/tex]
- Second monomial: [tex]\( ac \)[/tex]
4. [tex]\( 7c \times ab \)[/tex]:
- First monomial: [tex]\( 7c \)[/tex]
- Second monomial: [tex]\( ab \)[/tex]
5. [tex]\( 1 \times 7 a b c \)[/tex]:
- First monomial: [tex]\( 1 \)[/tex]
- Second monomial: [tex]\( 7 a b c \)[/tex]
So, the factor pairs for [tex]\( 7 a b c \)[/tex] are:
[tex]\[ (7, a bc), (7a, bc), (7b, ac), (7c, ab), (1, 7 a b c) \][/tex]
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