Factor completely 16a3b7 + 2a6b4 − 22a4b5.

Question

2(8a3b7 + a6b4 − 11a4b5)
2a3b4(8b3 + a3 − 11ab)
a3b4(16b3 + 2a3 − 22ab)
8b3 + a3 − 11ab

Answer 1

To factor the expression \( 16a^3b^7 + 2a^6b^4 - 22a^4b^5 \) completely, we can first look for a greatest common factor (GCF) among the terms.

1. Identify the GCF: The coefficients are 16, 2, and -22. The GCF of these coefficients is 2.

   Next, we consider the variables:

   • For \( a \): The smallest exponent is 3 (from \( a^3 \)).
   • For \( b \): The smallest exponent is 4 (from \( b^4 \)).

   Therefore, the GCF of the expression is \( 2a^3b^4 \).

2. Factor out the GCF: We factor out \( 2a^3b^4 \) from each term in the original expression:

   \[ 16a^3b^7 \div 2a^3b^4 = 8b^3 \]

   \[ 2a^6b^4 \div 2a^3b^4 = a^3 \]

   \[ -22a^4b^5 \div 2a^3b^4 = -11ab \]

   So after factoring out the GCF, we have:

   \[ 16a^3b^7 + 2a^6b^4 - 22a^4b^5 = 2a^3b^4(8b^3 + a^3 - 11ab) \]

Now, let's summarize the complete factored form:

\[ 2a^3b^4(8b^3 + a^3 - 11ab) \]

Looking at your choices, this corresponds to the form \( 2a^3b^4(8b^3 + a^3 - 11ab) \). Thus, the complete factorization is:

\[ \boxed{2a^3b^4(8b^3 + a^3 - 11ab)} \]