Certainly! Let's go through the process of factoring the given polynomial expression by finding the greatest common factor (GCF).
Given polynomial expression:
[tex]\[ 15a^4b^3 - 10a^2b^5 + 20a^5b^4 \][/tex]
### Step 1: Identify the coefficients and the variable parts.
The terms are:
1. [tex]\( 15a^4b^3 \)[/tex]
2. [tex]\( -10a^2b^5 \)[/tex]
3. [tex]\( 20a^5b^4 \)[/tex]
### Step 2: Find the greatest common factor (GCF) of the coefficients.
- The coefficients are 15, -10, and 20.
- The GCF of 15, -10, and 20 is 5.
### Step 3: Determine the GCF of the variable parts.
Look at the powers of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] in each term:
- For [tex]\( a \)[/tex]: The lowest power of [tex]\( a \)[/tex] present in all terms is [tex]\( a^2 \)[/tex].
- For [tex]\( b \)[/tex]: The lowest power of [tex]\( b \)[/tex] present in all terms is [tex]\( b^3 \)[/tex].
Therefore, the GCF of the variable part is [tex]\( a^2b^3 \)[/tex].
### Step 4: Combine the GCF of the coefficients and the variable parts.
Thus, the GCF of the whole expression is [tex]\( 5a^2b^3 \)[/tex].
### Step 5: Factor out the GCF from each term.
Now, we divide each term by the GCF [tex]\( 5a^2b^3 \)[/tex]:
[tex]\[ 15a^4b^3 \div 5a^2b^3 = 3a^2 \][/tex]
[tex]\[ -10a^2b^5 \div 5a^2b^3 = -2b^2 \][/tex]
[tex]\[ 20a^5b^4 \div 5a^2b^3 = 4a^3b \][/tex]
### Step 6: Write the original expression in factored form.
By factoring out [tex]\( 5a^2b^3 \)[/tex] from the original expression, we get:
[tex]\[ \begin{align*}
15a^4b^3 - 10a^2b^5 + 20a^5b^4 &= 5a^2b^3(3a^2) + 5a^2b^3(-2b^2) + 5a^2b^3(4a^3b) \\
&= 5a^2b^3(3a^2 - 2b^2 + 4a^3b)
\end{align*} \][/tex]
### Step 7: Simplify and present the final factored form.
The final factored form of the expression is:
[tex]\[ 5a^2b^3(4a^3b + 3a^2 - 2b^2) \][/tex]
Thus, the greatest common factor is [tex]\( 5a^2b^3 \)[/tex], and the factored form of the expression is:
[tex]\[ 5a^2b^3(4a^3b + 3a^2 - 2b^2) \][/tex]
