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Answer :
Sure, let's solve the given problem step by step.
We need to factor the trinomial [tex]\( 5x^3 + 45x^2 + 90x \)[/tex] completely.
### Step 1: Identify the Greatest Common Factor (GCF)
First, we will find the greatest common factor (GCF) of all the terms in the trinomial.
- The GCF of the coefficients [tex]\(5\)[/tex], [tex]\(45\)[/tex], and [tex]\(90\)[/tex] is [tex]\(5\)[/tex].
- Each term also includes the variable [tex]\(x\)[/tex], and the smallest power of [tex]\(x\)[/tex] in all terms is [tex]\(x\)[/tex].
So, the GCF of the trinomial is [tex]\( 5x \)[/tex].
### Step 2: Factor out the GCF
Next, we factor out [tex]\(5x\)[/tex] from each term:
[tex]\[
5x^3 + 45x^2 + 90x = 5x(x^2) + 5x(9x) + 5x(18) = 5x(x^2 + 9x + 18)
\][/tex]
### Step 3: Factor the Remaining Trinomial
Now, we need to factor the quadratic trinomial [tex]\(x^2 + 9x + 18\)[/tex]. We look for two numbers that multiply to [tex]\(18\)[/tex] (the constant term) and add up to [tex]\(9\)[/tex] (the coefficient of the middle term, [tex]\(x\)[/tex]).
Those two numbers are [tex]\(3\)[/tex] and [tex]\(6\)[/tex], since:
[tex]\[
3 \cdot 6 = 18 \quad \text{and} \quad 3 + 6 = 9
\][/tex]
Thus, we can factor [tex]\(x^2 + 9x + 18\)[/tex] as:
[tex]\[
x^2 + 9x + 18 = (x + 3)(x + 6)
\][/tex]
### Step 4: Combine the Factors
Now, combine this result with the GCF we factored out earlier.
[tex]\[
5x(x^2 + 9x + 18) = 5x(x + 3)(x + 6)
\][/tex]
Therefore, the trinomial [tex]\(5x^3 + 45x^2 + 90x\)[/tex] factors completely to:
[tex]\[
5x(x + 3)(x + 6)
\][/tex]
So, the final answer is:
A. [tex]\( 5x^3 + 45x^2 + 90x = 5x(x + 3)(x + 6) \)[/tex]
We need to factor the trinomial [tex]\( 5x^3 + 45x^2 + 90x \)[/tex] completely.
### Step 1: Identify the Greatest Common Factor (GCF)
First, we will find the greatest common factor (GCF) of all the terms in the trinomial.
- The GCF of the coefficients [tex]\(5\)[/tex], [tex]\(45\)[/tex], and [tex]\(90\)[/tex] is [tex]\(5\)[/tex].
- Each term also includes the variable [tex]\(x\)[/tex], and the smallest power of [tex]\(x\)[/tex] in all terms is [tex]\(x\)[/tex].
So, the GCF of the trinomial is [tex]\( 5x \)[/tex].
### Step 2: Factor out the GCF
Next, we factor out [tex]\(5x\)[/tex] from each term:
[tex]\[
5x^3 + 45x^2 + 90x = 5x(x^2) + 5x(9x) + 5x(18) = 5x(x^2 + 9x + 18)
\][/tex]
### Step 3: Factor the Remaining Trinomial
Now, we need to factor the quadratic trinomial [tex]\(x^2 + 9x + 18\)[/tex]. We look for two numbers that multiply to [tex]\(18\)[/tex] (the constant term) and add up to [tex]\(9\)[/tex] (the coefficient of the middle term, [tex]\(x\)[/tex]).
Those two numbers are [tex]\(3\)[/tex] and [tex]\(6\)[/tex], since:
[tex]\[
3 \cdot 6 = 18 \quad \text{and} \quad 3 + 6 = 9
\][/tex]
Thus, we can factor [tex]\(x^2 + 9x + 18\)[/tex] as:
[tex]\[
x^2 + 9x + 18 = (x + 3)(x + 6)
\][/tex]
### Step 4: Combine the Factors
Now, combine this result with the GCF we factored out earlier.
[tex]\[
5x(x^2 + 9x + 18) = 5x(x + 3)(x + 6)
\][/tex]
Therefore, the trinomial [tex]\(5x^3 + 45x^2 + 90x\)[/tex] factors completely to:
[tex]\[
5x(x + 3)(x + 6)
\][/tex]
So, the final answer is:
A. [tex]\( 5x^3 + 45x^2 + 90x = 5x(x + 3)(x + 6) \)[/tex]
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