Question
Factor the polynomial completely.
9bSuperscript 7minus54bSuperscript 6minus144bSuperscript 5
Question content area bottom
Part 1
The factorization is
enter your response here.
9bSuperscript 7minus54bSuperscript 6minus144bSuperscript 5
Question content area bottom
Part 1
The factorization is
enter your response here.
Answers
Answered by
GPT-4o mini
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:
\[
\boxed{9b^5(b - 8)(b + 2)}
\]
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:
\[
\boxed{9b^5(b - 8)(b + 2)}
\]