Answer :
Certainly! Let's go through the process of factorizing each expression step-by-step.
### (a) [tex]\( 15y + 5x \)[/tex]
Step 1: Find the greatest common factor (GCF) of the terms.
- The GCF of [tex]\( 15 \)[/tex] and [tex]\( 5 \)[/tex] is [tex]\( 5 \)[/tex].
Step 2: Factor out the GCF.
[tex]\[ 15y + 5x = 5(3y + x) \][/tex]
### Result:
[tex]\[ 15y + 5x = 5(x + 3y) \][/tex]
### (b) [tex]\( 21p - 7 \)[/tex]
Step 1: Find the GCF of the terms.
- The GCF of [tex]\( 21 \)[/tex] and [tex]\( 7 \)[/tex] is [tex]\( 7 \)[/tex].
Step 2: Factor out the GCF.
[tex]\[ 21p - 7 = 7(3p - 1) \][/tex]
### Result:
[tex]\[ 21p - 7 = 7(3p - 1) \][/tex]
### (d) [tex]\( 9ab - 12ac + 15bc \)[/tex]
Step 1: Factor out the common terms.
- Identify common factors in pairs:
- [tex]\( 9ab - 12ac \)[/tex] has a common factor of [tex]\( 3a \)[/tex]
- [tex]\( 15bc \)[/tex] does not share [tex]\( a \)[/tex], but [tex]\( b \)[/tex] and [tex]\( c \)[/tex] are present in other terms.
Step 2: Factor out the GCF from each pair of terms.
[tex]\[ 9ab - 12ac + 15bc = 3a(3b - 4c) + 15bc \][/tex]
This form can then be combined under a common factor:
[tex]\[ 9ab - 12ac + 15bc = 3a(3b - 4c) + 3b(5c) \][/tex]
This leads to:
[tex]\[ 9ab - 12ac + 15bc = 3(3ab - 4ac + 5bc) \][/tex]
### Result:
[tex]\[ 9ab - 12ac + 15bc = 3(3ab - 4ac + 5bc) \][/tex]
### (c) [tex]\( 60hg - 4 \)[/tex]
Step 1: Find the GCF of the terms.
- The GCF of [tex]\( 60 \)[/tex] and [tex]\( 4 \)[/tex] is [tex]\( 4 \)[/tex].
Step 2: Factor out the GCF.
[tex]\[ 60hg - 4 = 4(15hg - 1) \][/tex]
### Result:
[tex]\[ 60hg - 4 = 4(15hg - 1) \][/tex]
### (g) [tex]\( \frac{1}{5}ab - \frac{1}{5}ac \)[/tex]
Step 1: Find the GCF of the terms.
- The GCF of [tex]\( \frac{1}{5}ab \)[/tex] and [tex]\( \frac{1}{5}ac \)[/tex] is [tex]\( \frac{1}{5}a \)[/tex].
Step 2: Factor out the GCF.
[tex]\[ \frac{1}{5}ab - \frac{1}{5}ac = \frac{1}{5}a(b - c) \][/tex]
Since multiplying by a fraction is the same as dividing, the factor can be adjusted:
[tex]\[ \frac{1}{5}(ab - ac) = 0.2a(b - c) \][/tex]
### Result:
[tex]\[ \frac{1}{5}ab - \frac{1}{5}ac = 0.2a(b - c) \][/tex]
### (h) [tex]\( \frac{5}{2} - \frac{1}{4} \)[/tex]
Step 1: Find a common denominator.
- The common denominator for [tex]\( 2 \)[/tex] and [tex]\( 4 \)[/tex] is [tex]\( 4 \)[/tex].
Step 2: Convert each fraction to have the common denominator:
[tex]\[ \frac{5}{2} = \frac{10}{4} \][/tex]
[tex]\[ \frac{1}{4} = \frac{1}{4} \][/tex]
Step 3: Subtract the fractions:
[tex]\[ \frac{10}{4} - \frac{1}{4} = \frac{9}{4} \][/tex]
Step 4: Simplify the result:
[tex]\[ \frac{9}{4} = 2.25 \][/tex]
### Result:
[tex]\[ \frac{5}{2} - \frac{1}{4} = 2.25 \][/tex]
Thus, the complete factorization and simplification steps yield:
1. [tex]\( 15y + 5x = 5(x + 3y) \)[/tex]
2. [tex]\( 21p - 7 = 7(3p - 1) \)[/tex]
4. [tex]\( 9ab - 12ac + 15bc = 3(3ab - 4ac + 5bc) \)[/tex]
3. [tex]\( 60hg - 4 = 4(15hg - 1) \)[/tex]
5. [tex]\( \frac{1}{5}ab - \frac{1}{5}ac = 0.2a(b - c) \)[/tex]
6. [tex]\( \frac{5}{2} - \frac{1}{4} = 2.25 \)[/tex]
### (a) [tex]\( 15y + 5x \)[/tex]
Step 1: Find the greatest common factor (GCF) of the terms.
- The GCF of [tex]\( 15 \)[/tex] and [tex]\( 5 \)[/tex] is [tex]\( 5 \)[/tex].
Step 2: Factor out the GCF.
[tex]\[ 15y + 5x = 5(3y + x) \][/tex]
### Result:
[tex]\[ 15y + 5x = 5(x + 3y) \][/tex]
### (b) [tex]\( 21p - 7 \)[/tex]
Step 1: Find the GCF of the terms.
- The GCF of [tex]\( 21 \)[/tex] and [tex]\( 7 \)[/tex] is [tex]\( 7 \)[/tex].
Step 2: Factor out the GCF.
[tex]\[ 21p - 7 = 7(3p - 1) \][/tex]
### Result:
[tex]\[ 21p - 7 = 7(3p - 1) \][/tex]
### (d) [tex]\( 9ab - 12ac + 15bc \)[/tex]
Step 1: Factor out the common terms.
- Identify common factors in pairs:
- [tex]\( 9ab - 12ac \)[/tex] has a common factor of [tex]\( 3a \)[/tex]
- [tex]\( 15bc \)[/tex] does not share [tex]\( a \)[/tex], but [tex]\( b \)[/tex] and [tex]\( c \)[/tex] are present in other terms.
Step 2: Factor out the GCF from each pair of terms.
[tex]\[ 9ab - 12ac + 15bc = 3a(3b - 4c) + 15bc \][/tex]
This form can then be combined under a common factor:
[tex]\[ 9ab - 12ac + 15bc = 3a(3b - 4c) + 3b(5c) \][/tex]
This leads to:
[tex]\[ 9ab - 12ac + 15bc = 3(3ab - 4ac + 5bc) \][/tex]
### Result:
[tex]\[ 9ab - 12ac + 15bc = 3(3ab - 4ac + 5bc) \][/tex]
### (c) [tex]\( 60hg - 4 \)[/tex]
Step 1: Find the GCF of the terms.
- The GCF of [tex]\( 60 \)[/tex] and [tex]\( 4 \)[/tex] is [tex]\( 4 \)[/tex].
Step 2: Factor out the GCF.
[tex]\[ 60hg - 4 = 4(15hg - 1) \][/tex]
### Result:
[tex]\[ 60hg - 4 = 4(15hg - 1) \][/tex]
### (g) [tex]\( \frac{1}{5}ab - \frac{1}{5}ac \)[/tex]
Step 1: Find the GCF of the terms.
- The GCF of [tex]\( \frac{1}{5}ab \)[/tex] and [tex]\( \frac{1}{5}ac \)[/tex] is [tex]\( \frac{1}{5}a \)[/tex].
Step 2: Factor out the GCF.
[tex]\[ \frac{1}{5}ab - \frac{1}{5}ac = \frac{1}{5}a(b - c) \][/tex]
Since multiplying by a fraction is the same as dividing, the factor can be adjusted:
[tex]\[ \frac{1}{5}(ab - ac) = 0.2a(b - c) \][/tex]
### Result:
[tex]\[ \frac{1}{5}ab - \frac{1}{5}ac = 0.2a(b - c) \][/tex]
### (h) [tex]\( \frac{5}{2} - \frac{1}{4} \)[/tex]
Step 1: Find a common denominator.
- The common denominator for [tex]\( 2 \)[/tex] and [tex]\( 4 \)[/tex] is [tex]\( 4 \)[/tex].
Step 2: Convert each fraction to have the common denominator:
[tex]\[ \frac{5}{2} = \frac{10}{4} \][/tex]
[tex]\[ \frac{1}{4} = \frac{1}{4} \][/tex]
Step 3: Subtract the fractions:
[tex]\[ \frac{10}{4} - \frac{1}{4} = \frac{9}{4} \][/tex]
Step 4: Simplify the result:
[tex]\[ \frac{9}{4} = 2.25 \][/tex]
### Result:
[tex]\[ \frac{5}{2} - \frac{1}{4} = 2.25 \][/tex]
Thus, the complete factorization and simplification steps yield:
1. [tex]\( 15y + 5x = 5(x + 3y) \)[/tex]
2. [tex]\( 21p - 7 = 7(3p - 1) \)[/tex]
4. [tex]\( 9ab - 12ac + 15bc = 3(3ab - 4ac + 5bc) \)[/tex]
3. [tex]\( 60hg - 4 = 4(15hg - 1) \)[/tex]
5. [tex]\( \frac{1}{5}ab - \frac{1}{5}ac = 0.2a(b - c) \)[/tex]
6. [tex]\( \frac{5}{2} - \frac{1}{4} = 2.25 \)[/tex]
