Question 93733: Hello,
Can you PLEASE help me?? I cannot understand how to do this! I really do appreciate all your help!!
Find two integers c (positive or negative) for which each polynomial can be factored.
x^2+x+c
x^2-2x+c
x^2-3x+c
Thank you in advance!!
Found 2 solutions by stanbon, Edwin McCravy:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Find two integers c (positive or negative) for which each polynomial can be factored.
x^2+x+c
The coefficient of the middle term is "1".
You need to think of two numbers whose sum is "1"
Like: +2 and -1 whose product is -2
Like: +3 and -2 whose product is -6
Like: +4 and -3 whose product is -12
etc.
c could now be -2 or -6 or -12 etc.
---------
Try some of these and see that you do get a factorable trinomial.
--------------------------
x^2-2x+c
The coefficient of the middle term is "-2"
You need to think of two number whose sum is -2
like -3 and 1 whose product is -3
Like -4 and 2 whose product is -8
etc.
======================
x^2-3x+c
Like -4 and 1 whose product is -----
Like -5 and 2 whose product is ....
=================
Cheers,
Stan H.
Answer by Edwin McCravy(20055)  (Show Source):
You can put this solution on YOUR website!
Hello,
Can you PLEASE help me?? I cannot understand how to do this! I
really do appreciate all your help!!
Find two integers c (positive or negative) for which each
polynomial can be factored.
x² + x + c
For emphasis I will put a 1 before the x:
x² + 1x + c
That would factor if c were such that we could think of two
integers which have product c and sum 1.
So, for example, we think of two integers which have sum 1,
say +6 and -5, and then if c were their product, -30, it
would factor. So choose c as -30.
Checking:
x² + 1x - 30 factors as
(x + 6)(x - 5)
We could have picked instead two other integers
whose sum is +1. For instance we could have picked
-1 and +2, then c = (-1)(+2) = -2.
So another answer is c = -2. Checking,
x² + x - 2 factors as
(x - 1)(x + 2)
There are infinitely many answers for c.
----------------------------
x² - 2x + c
We do it the exact same way.
That would factor if c were such that we could think of two
integers which have product c and sum -2. So we think of two
integers which have sum -2, say -8 and +6, and then if c were
their product, -48, it would factor. So choose c as -48.
Checking:
x² - 2x - 48 factors as
(x - 8)(x + 6)
We could have picked instead two other integers
whose sum is -2. For instance we could have picked
+5 and -7, then c = (+5)(-7) = -35.
So another answer is c = -35. Checking,
x² + x - 35 factors as
(x + 5)(x - 7)
As in all such problems, there are infinitely many
answers for c.
===============================================
x² - 3x + c
I'll leave that one for you to do. Just think of two
numbers that have sum -3, say -2 and -1, or -8 and +5,
or -53 and +50, or -4 and +1, or -103 and 100,
or -999999999 and +999999996, J or .......
Then multiply them together to find a value for c.
Edwin
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