how do you factor the polynomial x^2+3x-4

Question

I don't understand how to factor polynomials.. ex: x^2+3-4

Answer 1

Factoring a polynomial of the form ax² + bx + c is the process of manipulating the previous expression so that its new form is something like (_x + _)(_x + _), where your job is to fill in the blanks.  This may look confusing at first, but a numerical example will help.  In the polynomial presented in the question, x² + 3x - 4 can be factored into (x + 4)(x - 1).  First let's double check that this is correct by carrying out the FOIL process (First, Outer, Inner, Last):  x*x - x*1 + 4x - 4*1 = x² +3x - 4.  

Ok, well where in the world did that come from?  Let's now examine the process of actually finding the factored form of the polynomial.  The key step is to identify the factors of the coefficient of x² and the constant term (the one with no x).  In this case, the coefficient of x² is 1, so our only factor is 1.  It is helpful to list the factors in pairs: the factors of 4 are 1,4, and 2,2.  Since 4 is subtracted instead of added, one of the factors of 4 in either pair must be negative.  What we're searching for is a combination of factors of 1 and 4 that add to 3 after we multiply each member of one pair by one of the other numbers in the other pair.  At this point it's guess and check.  Remember that one of the factors of 4 must be negative.  We can try the pair 1,1 and 2,-2, in which case we'd get 1*2 - 1*2 = 0.  This clearly didn't work.  We can try the pair 1,1 and 1,-4, in which case we'd get 1*1 - 1*4 = -3.  This didn't work either, but it looks like we're close.  Let's now try 1,1 and -1,4.  In this case we get -1*1 + 1*4 = 3.  Voila!  Now we know we need to write (x + 4)(x - 1).  

 

A final note:  if the coefficient of x is not 1, as in the above example, the process is no different in principle, there are just more pairs of factors to try.    

Answer 2

first remember that a= coefficent of x² b= is the coefficient of your x term and c= is your non algebraic term.  

hence using your example a=1 b=3 and c=-4

In order to factorize this figure out which numbers can multiply to get the coefficient a. that would be 1x1=1

then say which 2 numbers can multiply to get c but also cross multiple with your a factors to give numbers that will add up to your b term.

the numbers that could do this is -1 and 4 because -1x4 =-4 but u can also say -1x1=-1 and 1x4=4 and add them up you get +3 which is your b term.  

then take your first factor of a and your first factor of c which would be 1 and -1 the first factor of a would become x so you now have (x-1) 

then take second factor of a and second factor of c and you have (x+4) 

remember your a factors become your Xs 

answer is therefore (x-1) (x+4)

 

Answer 3

Factoring is the opposite of FOILing.  If you look at the factors (x+2)(x+5) and FOIL it (First, Outer, Inner, Last) you will end up with x²+5x+2x+10 which simplifies to x²+7x+10.  In your case, they give you the polynomial FOILed out and you have to put it back in factors.  In the example I wrote, you will notice that the 2 numbers in the factors (2 and 5) add to 7 and multiply to 10.  So for your example you have to find 2 numbers that multiply to -4 and add to 3. You will see that the only numbers that fit this criteria are -1 and 4.  -1•4=-4 and -1+4=3 so you have found the numbers you need and just need to plug them into the factors so your final answer is (x-1)(x+4).  To check your answer you can FOIL and you will get x²+3x-4  :)