SOLUTION: 4w^2+19w-5

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Question 182041This question is from textbook advanced mathematical concepts
: 4w^2+19w-5 This question is from textbook advanced mathematical concepts

Answer by jim_thompson5910(35256) About Me  (Show Source):
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Looking at the expression 4w%5E2%2B19w-5, we can see that the first coefficient is 4, the second coefficient is 19, and the last term is -5.


Now multiply the first coefficient 4 by the last term -5 to get %284%29%28-5%29=-20.


Now the question is: what two whole numbers multiply to -20 (the previous product) and add to the second coefficient 19?


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 19:


First NumberSecond NumberSum
1-201+(-20)=-19
2-102+(-10)=-8
4-54+(-5)=-1
-120-1+20=19
-210-2+10=8
-45-4+5=1



From the table, we can see that the two numbers -1 and 20 add to 19 (the middle coefficient).


So the two numbers -1 and 20 both multiply to -20 and add to 19


Now replace the middle term 19w with -w%2B20w. Remember, -1 and 20 add to 19. So this shows us that -w%2B20w=19w.


4w%5E2%2Bhighlight%28-w%2B20w%29-5 Replace the second term 19w with -w%2B20w.


%284w%5E2-w%29%2B%2820w-5%29 Group the terms into two pairs.


w%284w-1%29%2B%2820w-5%29 Factor out the GCF w from the first group.


w%284w-1%29%2B5%284w-1%29 Factor out 5 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.


%28w%2B5%29%284w-1%29 Combine like terms. Or factor out the common term 4w-1

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Answer:


So 4w%5E2%2B19w-5 factors to %28w%2B5%29%284w-1%29.


Note: you can check the answer by FOILing %28w%2B5%29%284w-1%29 to get 4w%5E2%2B19w-5 or by graphing the original expression and the answer (the two graphs should be identical).