Answer :
To factor the polynomial [tex]\(20x^3y + 15x^3 - 15x^4y\)[/tex], we need to find the greatest common factor (GCF) of all the terms and factor it out.
### Step 1: Identify the GCF of the coefficients
The numerical coefficients of the terms are 20, 15, and -15. The GCF of these numbers is 5.
### Step 2: Identify the GCF of the variables
The terms have the following variables:
- [tex]\(20x^3y\)[/tex] has variables [tex]\(x^3y\)[/tex].
- [tex]\(15x^3\)[/tex] has variables [tex]\(x^3\)[/tex].
- [tex]\(-15x^4y\)[/tex] has variables [tex]\(x^4y\)[/tex].
The smallest power of [tex]\(x\)[/tex] present in all terms is [tex]\(x^3\)[/tex]. Both terms [tex]\(20x^3y\)[/tex] and [tex]\(-15x^4y\)[/tex] have [tex]\(y\)[/tex], but the term [tex]\(15x^3\)[/tex] does not have [tex]\(y\)[/tex], so we do not include [tex]\(y\)[/tex] in the GCF. Therefore, the GCF of the variables is [tex]\(x^3\)[/tex].
### Step 3: Combine to find the overall GCF
The overall GCF of the polynomial is [tex]\(5x^3\)[/tex].
### Step 4: Factor the GCF out of each term
Now we factor out [tex]\(5x^3\)[/tex] from each term of the polynomial:
- From [tex]\(20x^3y\)[/tex]: [tex]\(20x^3y \div 5x^3 = 4y\)[/tex]
- From [tex]\(15x^3\)[/tex]: [tex]\(15x^3 \div 5x^3 = 3\)[/tex]
- From [tex]\(-15x^4y\)[/tex]: [tex]\(-15x^4y \div 5x^3 = -3xy\)[/tex]
### Step 5: Write the factored form
After factoring out the GCF [tex]\(5x^3\)[/tex], the polynomial becomes:
[tex]\[ 5x^3(4y + 3 - 3xy) \][/tex]
So, the factored form of the polynomial [tex]\(20x^3y + 15x^3 - 15x^4y\)[/tex] is [tex]\(\boxed{5x^3(4y + 3 - 3xy)}\)[/tex].
### Step 1: Identify the GCF of the coefficients
The numerical coefficients of the terms are 20, 15, and -15. The GCF of these numbers is 5.
### Step 2: Identify the GCF of the variables
The terms have the following variables:
- [tex]\(20x^3y\)[/tex] has variables [tex]\(x^3y\)[/tex].
- [tex]\(15x^3\)[/tex] has variables [tex]\(x^3\)[/tex].
- [tex]\(-15x^4y\)[/tex] has variables [tex]\(x^4y\)[/tex].
The smallest power of [tex]\(x\)[/tex] present in all terms is [tex]\(x^3\)[/tex]. Both terms [tex]\(20x^3y\)[/tex] and [tex]\(-15x^4y\)[/tex] have [tex]\(y\)[/tex], but the term [tex]\(15x^3\)[/tex] does not have [tex]\(y\)[/tex], so we do not include [tex]\(y\)[/tex] in the GCF. Therefore, the GCF of the variables is [tex]\(x^3\)[/tex].
### Step 3: Combine to find the overall GCF
The overall GCF of the polynomial is [tex]\(5x^3\)[/tex].
### Step 4: Factor the GCF out of each term
Now we factor out [tex]\(5x^3\)[/tex] from each term of the polynomial:
- From [tex]\(20x^3y\)[/tex]: [tex]\(20x^3y \div 5x^3 = 4y\)[/tex]
- From [tex]\(15x^3\)[/tex]: [tex]\(15x^3 \div 5x^3 = 3\)[/tex]
- From [tex]\(-15x^4y\)[/tex]: [tex]\(-15x^4y \div 5x^3 = -3xy\)[/tex]
### Step 5: Write the factored form
After factoring out the GCF [tex]\(5x^3\)[/tex], the polynomial becomes:
[tex]\[ 5x^3(4y + 3 - 3xy) \][/tex]
So, the factored form of the polynomial [tex]\(20x^3y + 15x^3 - 15x^4y\)[/tex] is [tex]\(\boxed{5x^3(4y + 3 - 3xy)}\)[/tex].
