To factor the expression [tex]\( 12a^3b + 8a^2b^2 - 20ab^3 \)[/tex] completely, let's follow a step-by-step procedure:
1. Identify the Greatest Common Factor (GCF):
- First, list the coefficients: [tex]\( 12, 8, 20 \)[/tex].
- Find the GCF of these coefficients. The GCF of [tex]\( 12, 8, \)[/tex] and [tex]\( 20 \)[/tex] is [tex]\( 4 \)[/tex].
- For the variables, determine the lowest power of each variable:
- For [tex]\( a \)[/tex], the lowest power is [tex]\( a \)[/tex].
- For [tex]\( b \)[/tex], the lowest power is [tex]\( b \)[/tex].
- So, the GCF of the whole expression is [tex]\( 4ab \)[/tex].
2. Factor out the GCF from each term:
- [tex]\( \frac{12a^3b}{4ab} = 3a^2 \)[/tex]
- [tex]\( \frac{8a^2b^2}{4ab} = 2ab \)[/tex]
- [tex]\( \frac{20ab^3}{4ab} = 5b^2 \)[/tex]
3. Write the expression factored by the GCF:
[tex]\[
12a^3b + 8a^2b^2 - 20ab^3 = 4ab(3a^2 + 2ab - 5b^2)
\][/tex]
4. Further Factor the Polynomial Inside the Parentheses:
- Notice that [tex]\( 3a^2 + 2ab - 5b^2 \)[/tex] can be further factored into products of binomials.
[tex]\[
3a^2 + 2ab - 5b^2 = (a - b)(3a + 5b)
\][/tex]
5. Combine the Factored Form:
- Substitute back the factored polynomial:
[tex]\[
4ab(3a^2 + 2ab - 5b^2) = 4ab(a - b)(3a + 5b)
\][/tex]
Therefore, the completely factored form of the expression [tex]\( 12a^3b + 8a^2b^2 - 20ab^3 \)[/tex] is:
[tex]\[
4ab(a - b)(3a + 5b)
\][/tex]
This detailed process matches our final factored result.
