Answer :
Sure, let's walk through each of these steps one by one:
### Step 1: Factoring [tex]\(a(x + y + z)\)[/tex]
We start by factoring the expression [tex]\(a(x + y + z)\)[/tex]. Factoring means to express the expression as a product of its factors.
Given Expression: [tex]\(a(x + y + z)\)[/tex]
Factored Form:
Since [tex]\(a\)[/tex] is a common factor in each term of the expression, we can factor out [tex]\(a\)[/tex]:
[tex]\[ a(x + y + z) \][/tex]
So, the factored form is:
[tex]\[ a(x + y + z) \][/tex]
### Step 2: Simplify the Expression
[tex]\[ \frac{\frac{a}{x}+\frac{a}{y}+\frac{a}{z}}{a(x+y+z)} \][/tex]
Let's simplify this expression step by step:
Given Expression:
[tex]\[ \frac{\frac{a}{x}+\frac{a}{y}+\frac{a}{z}}{a(x+y+z)} \][/tex]
1. Combine the numerator:
[tex]\[ \frac{a}{x}+\frac{a}{y}+\frac{a}{z} = a\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) \][/tex]
2. Rewrite the original expression:
[tex]\[ \frac{a\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right)}{a(x+y+z)} \][/tex]
3. Cancel out the common factor [tex]\(a\)[/tex] from the numerator and the denominator:
[tex]\[ \frac{\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right)}{x+y+z} \][/tex]
4. Express the simplified form:
[tex]\[ \frac{x y + x z + y z}{x y z (x + y + z)} \][/tex]
So, the simplified form is:
[tex]\[ \frac{x y + x z + y z}{x y z (x + y + z)} \][/tex]
### Step 3: Factor [tex]\(4x^3 + 8x^2 + 12x\)[/tex]
We factor out the greatest common factor (GCF) from the expression:
Given Expression: [tex]\(4x^3 + 8x^2 + 12x\)[/tex]
1. Identify the GCF: The GCF is [tex]\(4x\)[/tex].
2. Factor out the GCF:
[tex]\[ 4x^3 + 8x^2 + 12x = 4x(x^2 + 2x + 3) \][/tex]
So, the factored form is:
[tex]\[ 4x(x^2 + 2x + 3) \][/tex]
### Step 4: Simplify the Fraction
[tex]\[ \frac{x+x+x}{x+x+x+4} \][/tex]
First, let's simplify the numerator and the denominator separately:
Given Fraction:
[tex]\[ \frac{x+x+x}{x+x+x+4} \][/tex]
1. Combine like terms:
[tex]\[ \frac{3x}{3x + 4} \][/tex]
So, the simplified fraction is:
[tex]\[ \frac{3x}{3x + 4} \][/tex]
### Step 5: Factor [tex]\(6x + 18y - 9z\)[/tex]
We factor out the greatest common factor (GCF):
Given Expression: [tex]\(6x + 18y - 9z\)[/tex]
1. Identify the GCF: The GCF is [tex]\(3\)[/tex].
2. Factor out the GCF:
[tex]\[ 6x + 18y - 9z = 3(2x + 6y - 3z) \][/tex]
So, the factored form is:
[tex]\[ 3(2x + 6y - 3z) \][/tex]
### Step 6: Factor [tex]\(-3a^3 + 6a^2 + 12\)[/tex]
We factor out the greatest common factor (GCF):
Given Expression: [tex]\(-3a^3 + 6a^2 + 12\)[/tex]
1. Identify the GCF: The GCF is [tex]\(-3\)[/tex].
2. Factor out the GCF:
[tex]\[ -3a^3 + 6a^2 + 12 = -3(a^3 - 2a^2 - 4) \][/tex]
So, the factored form is:
[tex]\[ -3(a^3 - 2a^2 - 4) \][/tex]
These are the complete steps and detailed solutions for the given expressions.
### Step 1: Factoring [tex]\(a(x + y + z)\)[/tex]
We start by factoring the expression [tex]\(a(x + y + z)\)[/tex]. Factoring means to express the expression as a product of its factors.
Given Expression: [tex]\(a(x + y + z)\)[/tex]
Factored Form:
Since [tex]\(a\)[/tex] is a common factor in each term of the expression, we can factor out [tex]\(a\)[/tex]:
[tex]\[ a(x + y + z) \][/tex]
So, the factored form is:
[tex]\[ a(x + y + z) \][/tex]
### Step 2: Simplify the Expression
[tex]\[ \frac{\frac{a}{x}+\frac{a}{y}+\frac{a}{z}}{a(x+y+z)} \][/tex]
Let's simplify this expression step by step:
Given Expression:
[tex]\[ \frac{\frac{a}{x}+\frac{a}{y}+\frac{a}{z}}{a(x+y+z)} \][/tex]
1. Combine the numerator:
[tex]\[ \frac{a}{x}+\frac{a}{y}+\frac{a}{z} = a\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) \][/tex]
2. Rewrite the original expression:
[tex]\[ \frac{a\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right)}{a(x+y+z)} \][/tex]
3. Cancel out the common factor [tex]\(a\)[/tex] from the numerator and the denominator:
[tex]\[ \frac{\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right)}{x+y+z} \][/tex]
4. Express the simplified form:
[tex]\[ \frac{x y + x z + y z}{x y z (x + y + z)} \][/tex]
So, the simplified form is:
[tex]\[ \frac{x y + x z + y z}{x y z (x + y + z)} \][/tex]
### Step 3: Factor [tex]\(4x^3 + 8x^2 + 12x\)[/tex]
We factor out the greatest common factor (GCF) from the expression:
Given Expression: [tex]\(4x^3 + 8x^2 + 12x\)[/tex]
1. Identify the GCF: The GCF is [tex]\(4x\)[/tex].
2. Factor out the GCF:
[tex]\[ 4x^3 + 8x^2 + 12x = 4x(x^2 + 2x + 3) \][/tex]
So, the factored form is:
[tex]\[ 4x(x^2 + 2x + 3) \][/tex]
### Step 4: Simplify the Fraction
[tex]\[ \frac{x+x+x}{x+x+x+4} \][/tex]
First, let's simplify the numerator and the denominator separately:
Given Fraction:
[tex]\[ \frac{x+x+x}{x+x+x+4} \][/tex]
1. Combine like terms:
[tex]\[ \frac{3x}{3x + 4} \][/tex]
So, the simplified fraction is:
[tex]\[ \frac{3x}{3x + 4} \][/tex]
### Step 5: Factor [tex]\(6x + 18y - 9z\)[/tex]
We factor out the greatest common factor (GCF):
Given Expression: [tex]\(6x + 18y - 9z\)[/tex]
1. Identify the GCF: The GCF is [tex]\(3\)[/tex].
2. Factor out the GCF:
[tex]\[ 6x + 18y - 9z = 3(2x + 6y - 3z) \][/tex]
So, the factored form is:
[tex]\[ 3(2x + 6y - 3z) \][/tex]
### Step 6: Factor [tex]\(-3a^3 + 6a^2 + 12\)[/tex]
We factor out the greatest common factor (GCF):
Given Expression: [tex]\(-3a^3 + 6a^2 + 12\)[/tex]
1. Identify the GCF: The GCF is [tex]\(-3\)[/tex].
2. Factor out the GCF:
[tex]\[ -3a^3 + 6a^2 + 12 = -3(a^3 - 2a^2 - 4) \][/tex]
So, the factored form is:
[tex]\[ -3(a^3 - 2a^2 - 4) \][/tex]
These are the complete steps and detailed solutions for the given expressions.
