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how do i factor the trinomial r^2-r-56?

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2 Answers

In factoring a trinomial, we know that the terms will appear as:

(?r +/- ?)(?r +/- ?)

Because there iskno written coefficient in front of the r2, we understand that the coefficient is 1.  Since the only way to factor 1 is 1X1, we know the coefficient of each of the r's is (understood) 1.

(r +/- ?)(r +/- ?)

By looking at the sign on the last term (56), we can determine if the signs in the factors will be the same or opposite.  Since the sign on the last term is negative, one of the signs will be plus and the other will be minus.

Also, since the sign of the last term is negative, we know that we will be looking for the factors of the last term that produce a difference of 1 (the coefficient on the second term).

The factors of 56 are: 1X56, 2X28, 4X14, 7X8.

Since the difference of 7 and 8 is 1, those of the factors we need.

(r +/- 7)(r +/- 8)

Now, we need to determine which factor gets the + sign and which factor gets the - sign.  Since the sign on the second term (r) is negative, the larger number gets the negative.  8 is larger than 7, so 8 gets the negative sign.

(r + 7)(r - 8)



to factor this type of polynomial, you need to consider what it looks like.  First, look for any common factors, which this one doesn't have.  Then, you need to look at it's form.  This trinomial is a standard quadratic where you have: ax^2+bx+c, where a=1, b=-1, and c=-56.  From this, you need to find a pair of numbers that multiply to c and add to b.  You don't have to consider a here because it is 1.  Any other time, you would need to.  So, a pair that multiply to -56 and add to -1.  That pair is -8 and 7.  Once you have those, you create your binomial factors.  In this case, (r-8)(r+7).  If a were something other than 1, that would not be the next step.  Hope this helps you out.