To factor the expression [tex]\(50 a^2 b^5 - 35 a^4 b^3 + 5 a^3 b^4\)[/tex] completely, let's follow a step-by-step approach:
1. Identify the Greatest Common Factor (GCF):
We look for the largest factor that is common in each term.
- For the coefficients: the GCF of 50, 35, and 5 is 5.
- For the [tex]\(a\)[/tex] terms: the common factor is [tex]\(a^2\)[/tex].
- For the [tex]\(b\)[/tex] terms: the common factor is [tex]\(b^3\)[/tex].
Therefore, the GCF of the entire expression is [tex]\(5a^2b^3\)[/tex].
2. Factor out the GCF:
Let's factor [tex]\(5a^2b^3\)[/tex] out of each term in the expression:
[tex]\[
50 a^2 b^5 - 35 a^4 b^3 + 5 a^3 b^4 = 5 a^2 b^3 (10 b^2) - 5 a^2 b^3 (7 a^2) + 5 a^2 b^3 (a b)
\][/tex]
3. Simplify inside the parentheses:
After factoring out the GCF [tex]\(5a^2b^3\)[/tex], the expression inside the parentheses simplifies as:
[tex]\[
50 a^2 b^5 - 35 a^4 b^3 + 5 a^3 b^4 = 5a^2b^3 (10 b^2 - 7 a^2 + ab)
\][/tex]
4. Rewrite the final factored form:
Thus, the completely factored form of the expression [tex]\(50 a^2 b^5 - 35 a^4 b^3 + 5 a^3 b^4\)[/tex] is:
[tex]\[
50 a^2 b^5 - 35 a^4 b^3 + 5 a^3 b^4 = -5 a^2 b^3 (7 a^2 - ab - 10 b^2)
\][/tex]
Hence, the completely factored form of the given expression is:
[tex]\[
-5a^2b^3(7a^2 - ab - 10b^2).
\][/tex]
