Factor the polynomial completely.

9bSuperscript 7minus54bSuperscript 6minus144bSuperscript 5
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Part 1
The factorization is
  
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1 answer

To factor the polynomial \(9b^7 - 54b^6 - 144b^5\), we can start by factoring out the greatest common factor (GCF) from all the terms.

The GCF of the coefficients \(9\), \(-54\), and \(-144\) is \(9\). For the variable part, we can factor out \(b^5\) since it is the lowest power of \(b\) in the terms.

So, we factor out \(9b^5\):

\[ 9b^7 - 54b^6 - 144b^5 = 9b^5(b^2 - 6b - 16) \]

Next, we need to factor the quadratic \(b^2 - 6b - 16\). To do this, we can look for two numbers that multiply to \(-16\) (the constant term) and add to \(-6\) (the coefficient of the linear term).

The numbers that meet these conditions are \(-8\) and \(2\). Thus, we can factor the quadratic as:

\[ b^2 - 6b - 16 = (b - 8)(b + 2) \]

Putting it all together, we have:

\[ 9b^7 - 54b^6 - 144b^5 = 9b^5(b - 8)(b + 2) \]

The completely factored form of the polynomial is:

\[ \boxed{9b^5(b - 8)(b + 2)} \]