SOLUTION: factoring a quadratic with leading coefficient greater than one z^2+12z+20

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Question 91429: factoring a quadratic with leading coefficient greater than one
z^2+12z+20
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression z%5E2%2B12z%2B20, we can see that the first coefficient is 1, the second coefficient is 12, and the last term is 20.



Now multiply the first coefficient 1 by the last term 20 to get %281%29%2820%29=20.



Now the question is: what two whole numbers multiply to 20 (the previous product) and add to the second coefficient 12?



To find these two numbers, we need to list all of the factors of 20 (the previous product).



Factors of 20:

1,2,4,5,10,20

-1,-2,-4,-5,-10,-20



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to 20.

1*20 = 20
2*10 = 20
4*5 = 20
(-1)*(-20) = 20
(-2)*(-10) = 20
(-4)*(-5) = 20


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 12:



First NumberSecond NumberSum
1201+20=21
2102+10=12
454+5=9
-1-20-1+(-20)=-21
-2-10-2+(-10)=-12
-4-5-4+(-5)=-9




From the table, we can see that the two numbers 2 and 10 add to 12 (the middle coefficient).



So the two numbers 2 and 10 both multiply to 20 and add to 12



Now replace the middle term 12z with 2z%2B10z. Remember, 2 and 10 add to 12. So this shows us that 2z%2B10z=12z.



z%5E2%2Bhighlight%282z%2B10z%29%2B20 Replace the second term 12z with 2z%2B10z.



%28z%5E2%2B2z%29%2B%2810z%2B20%29 Group the terms into two pairs.



z%28z%2B2%29%2B%2810z%2B20%29 Factor out the GCF z from the first group.



z%28z%2B2%29%2B10%28z%2B2%29 Factor out 10 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



%28z%2B10%29%28z%2B2%29 Combine like terms. Or factor out the common term z%2B2



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Answer:



So z%5E2%2B12%2Az%2B20 factors to %28z%2B10%29%28z%2B2%29.



In other words, z%5E2%2B12%2Az%2B20=%28z%2B10%29%28z%2B2%29.



Note: you can check the answer by expanding %28z%2B10%29%28z%2B2%29 to get z%5E2%2B12%2Az%2B20 or by graphing the original expression and the answer (the two graphs should be identical).