Factor polynomial $$ 48b^{7}+4b^{6}+32b^{5} $$

$$ 48b^{7}+4b^{6}+32b^{5} = 4b^{5}( 12b^{2}+b+8 ) $$

Step 1 :

After factoring out $ 4b^{5} $ we have:

$$ 48b^{7}+4b^{6}+32b^{5} = 4b^{5} ( 12b^{2}+b+8 ) $$

Step 2 :

Step 2: Identify constants $ a $ , $ b $ and $ c $.
$ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant. In this case:

$$ a = 12 , b = 1 ~ \text{ and } ~ c = 8 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 12 }$ by the constant term $\color{blue}{c = 8} $.

$$ a \cdot c = 96 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 96 $ and add to $ b = 1 $.

Step 5: All pairs of numbers with a product of $ 96 $ are:

PRODUCT = 96
1     96	-1     -96
2     48	-2     -48
3     32	-3     -32
4     24	-4     -24
6     16	-6     -16
8     12	-8     -12

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 1 }$

Step 7: Because none of these pairs will give us a sum of $ \color{blue}{ 1 }$, we conclude the polynomial cannot be factored.

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