Answer :
Sure! Let's factor the expression [tex]\(45x^3 + 90x\)[/tex] using three different methods.
### Step 1: Factor out the Greatest Common Factor (GCF)
1. Identify the greatest common factor of the terms. The GCF of 45 and 90 is 45, and the GCF of the variables [tex]\(x^3\)[/tex] and [tex]\(x\)[/tex] is [tex]\(x\)[/tex].
2. Factor out 45x:
[tex]\[
45x^3 + 90x = 45x(x^2 + 2)
\][/tex]
### Step 2: Factor by Grouping
To factor by grouping, we'll rewrite the expression in a way that allows grouping of terms:
1. Rewrite the expression: [tex]\(45x^3 + 90x = 3(15x^3) + 6(15x)\)[/tex].
2. Find the GCF for each group:
- For [tex]\(15x^3\)[/tex] and [tex]\(15x\)[/tex], the GCF is [tex]\(15x\)[/tex].
3. Factor out the GCF from each group:
[tex]\[
= 15x(3x^2 + 6)
\][/tex]
### Step 3: Using Alternative Factorization
1. Notice the expression can be factored further in this way, given [tex]\(45x(x^2 + 2)\)[/tex]:
2. Split the constant factor, if needed, to let you explore other equivalent forms.
3. For instance, you can express 45 as [tex]\(3 \times 15\)[/tex], and refactor:
[tex]\[
= 3(15x)(x^2 + 2)
\][/tex]
So, the three different equivalent expressions we derived are:
1. [tex]\(45x(x^2 + 2)\)[/tex]
2. [tex]\(15x(3x^2 + 6)\)[/tex]
3. [tex]\(3(15x)(x^2 + 2)\)[/tex]
Each of these expressions is equivalent to the original expression [tex]\(45x^3 + 90x\)[/tex] but rewritten in different factored forms.
### Step 1: Factor out the Greatest Common Factor (GCF)
1. Identify the greatest common factor of the terms. The GCF of 45 and 90 is 45, and the GCF of the variables [tex]\(x^3\)[/tex] and [tex]\(x\)[/tex] is [tex]\(x\)[/tex].
2. Factor out 45x:
[tex]\[
45x^3 + 90x = 45x(x^2 + 2)
\][/tex]
### Step 2: Factor by Grouping
To factor by grouping, we'll rewrite the expression in a way that allows grouping of terms:
1. Rewrite the expression: [tex]\(45x^3 + 90x = 3(15x^3) + 6(15x)\)[/tex].
2. Find the GCF for each group:
- For [tex]\(15x^3\)[/tex] and [tex]\(15x\)[/tex], the GCF is [tex]\(15x\)[/tex].
3. Factor out the GCF from each group:
[tex]\[
= 15x(3x^2 + 6)
\][/tex]
### Step 3: Using Alternative Factorization
1. Notice the expression can be factored further in this way, given [tex]\(45x(x^2 + 2)\)[/tex]:
2. Split the constant factor, if needed, to let you explore other equivalent forms.
3. For instance, you can express 45 as [tex]\(3 \times 15\)[/tex], and refactor:
[tex]\[
= 3(15x)(x^2 + 2)
\][/tex]
So, the three different equivalent expressions we derived are:
1. [tex]\(45x(x^2 + 2)\)[/tex]
2. [tex]\(15x(3x^2 + 6)\)[/tex]
3. [tex]\(3(15x)(x^2 + 2)\)[/tex]
Each of these expressions is equivalent to the original expression [tex]\(45x^3 + 90x\)[/tex] but rewritten in different factored forms.
