SOLUTION: Level 1, General Polynomials, Degree 2 Write your answer in the form (ay + b) (cy +d) .... (ey +f) New Problem, Factor: - 4 y2 - y + 3

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Level 1, General Polynomials, Degree 2 Write your answer in the form (ay + b) (cy +d) .... (ey +f) New Problem, Factor: - 4 y2 - y + 3       Log On


   

Question 103799: Level 1, General Polynomials, Degree 2
Write your answer in the form (ay + b) (cy +d) .... (ey +f)
New Problem, Factor:

- 4 y2 - y + 3
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)


-4%2Ay%5E2-y%2B3 Start with the given expression.



-%284y%5E2%2By-3%29 Factor out the GCF -1.



Now let's try to factor the inner expression 4y%5E2%2By-3



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Looking at the expression 4y%5E2%2By-3, we can see that the first coefficient is 4, the second coefficient is 1, and the last term is -3.



Now multiply the first coefficient 4 by the last term -3 to get %284%29%28-3%29=-12.



Now the question is: what two whole numbers multiply to -12 (the previous product) and add to the second coefficient 1?



To find these two numbers, we need to list all of the factors of -12 (the previous product).



Factors of -12:

1,2,3,4,6,12

-1,-2,-3,-4,-6,-12



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



These factors pair up and multiply to -12.

1*(-12) = -12
2*(-6) = -12
3*(-4) = -12
(-1)*(12) = -12
(-2)*(6) = -12
(-3)*(4) = -12


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 1:



First NumberSecond NumberSum
1-121+(-12)=-11
2-62+(-6)=-4
3-43+(-4)=-1
-112-1+12=11
-26-2+6=4
-34-3+4=1




From the table, we can see that the two numbers -3 and 4 add to 1 (the middle coefficient).



So the two numbers -3 and 4 both multiply to -12 and add to 1



Now replace the middle term 1y with -3y%2B4y. Remember, -3 and 4 add to 1. So this shows us that -3y%2B4y=1y.



4y%5E2%2Bhighlight%28-3y%2B4y%29-3 Replace the second term 1y with -3y%2B4y.



%284y%5E2-3y%29%2B%284y-3%29 Group the terms into two pairs.



y%284y-3%29%2B%284y-3%29 Factor out the GCF y from the first group.



y%284y-3%29%2B1%284y-3%29 Factor out 1 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.



%28y%2B1%29%284y-3%29 Combine like terms. Or factor out the common term 4y-3



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So -1%284y%5E2%2By-3%29 then factors further to -%28y%2B1%29%284y-3%29



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Answer:



So -4%2Ay%5E2-y%2B3 completely factors to -%28y%2B1%29%284y-3%29.



In other words, -4%2Ay%5E2-y%2B3=-%28y%2B1%29%284y-3%29.



Note: you can check the answer by expanding -%28y%2B1%29%284y-3%29 to get -4%2Ay%5E2-y%2B3 or by graphing the original expression and the answer (the two graphs should be identical).