Factoring Pattern for x2 + bx+ c where c is negative 5-8
Goal- to factor quadratic trinomials whose quadratic coefficient is 1 and whose constant is negative
The method used in this lesson is very similar to that used to factor x2 +bx + c , c is positive, except instead of the sum of the two factors you find their difference.
Remember with x2 +bx + c , c is positive, you find two numbers whose product is c and whose sum is b.
Remember with x2 +bx + c , c is positive, you find two numbers whose product is c and whose sum is b.
This time find two numbers whose product is c (which is negative)—so ONE of the TWO factors must be negative. You will have either (x + )(x - ) or (x - )(x + ) Since c is negative, one of the two factors MUST be negative.
The first sign in x2 + bx + c , c is negative determines “Who wins!” Let’s rewrite x2 +bx + c , c is negative as either x2 +bx - c , or x2 -bx - c to see how this works.
We started with x2 –x – 20
Set up your hugs…. (x - ) (x + )… with the winning sign going in the first set of hugs
Set up your hugs…. (x - ) (x + )… with the winning sign going in the first set of hugs
Then using the X method find two numbers whose product is 20 and whose difference is 1 ( and in this case actually -1) We found 5 and 4 works and the 5 must be negative to get -1
so (x -5)(x + 4)
If the quadratic was x2+x-20 you would still have the same factors 5 and 4 but this time the difference is +1 so you would have ( x +5)(x -4)
How about x2 + 29a – 30
The factoring pattern is ( x + )(x - )
Use the X method and find two numbers whose product is 30 and whose difference is 29
We find it has to be 30 and 1 (x+30)(x -1)
Use the X method and find two numbers whose product is 30 and whose difference is 29
We find it has to be 30 and 1 (x+30)(x -1)
To check just FOIL, FireWorks, use the BOX method or just double-distribute to get back to where you started!!
x2-4kx +12k2
This time we have another variable on the last two terms so the factoring pattern starts out as
(x- _k)(x+ _k)
But again we just need to find two numbers whose product is 12 and the difference is 4
6 and 2 work so its ( x -6k)(x + 2k)
(x- _k)(x+ _k)
But again we just need to find two numbers whose product is 12 and the difference is 4
6 and 2 work so its ( x -6k)(x + 2k)
Find all the integral values for k for which the given polynomial can be factored.
c2-kc-20
For this exercise, set up a T chart with all the factors that multiply to 20
we found 1 and 20, 2 and 10, and 4 and 5. Now taking their differences, we find that ± 19 ± 8 ± 1 all work.
Find two negative values for k for which the given polynomial can be factored. (There are many possibilities—the class found several)
y2 + 4y + k
We found -5, -12, -77, 45, -21, … and ....
