Question 130497: Hi there, was wondering if someone could explain this for me?
Explain the meaning of "factor of a polynomial" and explain the meaning "to factor a polynomial"
thanks
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Remember, if you factor a number like 56, you simply break that number into it's product of prime numbers. So 56 factors to
56=2*2*2*7
So with polynomials, a similar thing occurs when you factor a polynomial
Looking at  we can see that the first term is  and the last term is  where the coefficients are 1 and 6 respectively.
Now multiply the first coefficient 1 and the last coefficient 6 to get 6. Now what two numbers multiply to 6 and add to the middle coefficient 5? Let's list all of the factors of 6:
Factors of 6:
1,2,3,6
-1,-2,-3,-6 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 6
1*6
2*3
(-1)*(-6)
(-2)*(-3)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to 5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 5
| First Number | Second Number | Sum | | 1 | 6 | 1+6=7 | | 2 | 3 | 2+3=5 | | -1 | -6 | -1+(-6)=-7 | | -2 | -3 | -2+(-3)=-5 |
From this list we can see that 2 and 3 add up to 5 and multiply to 6
Now looking at the expression , replace  with  (notice adds up to . So it is equivalent to )
Now let's factor  by grouping:
 Group like terms
 Factor out the GCF of  out of the first group. Factor out the GCF of  out of the second group
 Since we have a common term of  , we can combine like terms
So factors to
Looking at the factorization, the terms  and are the factors of the polynomial .
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