Factor out the GCF from the following polynomial.

Question

32 m Superscript 7 Baseline minus 96 m Superscript 6 Baseline plus 144 m Superscript 5 Baseline minus 104 m cubed
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Part 1
32 m Superscript 7 Baseline minus 96 m Superscript 6 Baseline plus 144 m Superscript 5 Baseline minus 104 m cubedequals
  
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Answer 1

To factor out the greatest common factor (GCF) from the polynomial \(32m^7 - 96m^6 + 144m^5 - 104m^3\), we first look for the GCF of the coefficients (32, 96, 144, 104) and then consider the variable part.

1. Finding the GCF of the coefficients:

   • The coefficients are 32, 96, 144, and 104.
   • The GCF of these numbers can be found by breaking them down into their prime factorizations:
     • \(32 = 2^5\)
     • \(96 = 2^5 \times 3\)
     • \(144 = 2^4 \times 3^2\)
     • \(104 = 2^3 \times 13\)
   • The smallest power of 2 among these is \(2^3\). Hence, the GCF for the coefficients is \(8\).

2. Finding the GCF of the variable parts:

   • The variable parts are \(m^7, m^6, m^5, m^3\).
   • The smallest power of \(m\) is \(m^3\).

3. Combining the results:

   • Therefore, the GCF of the polynomial is \(8m^3\).

4. Factoring out the GCF:

   • Now we can factor out \(8m^3\) from each term: \[ 32m^7 - 96m^6 + 144m^5 - 104m^3 = 8m^3(4m^4 - 12m^3 + 18m^2 - 13). \]

So the final factored form is: \[ 32m^7 - 96m^6 + 144m^5 - 104m^3 = 8m^3(4m^4 - 12m^3 + 18m^2 - 13). \]