completely factor the following expressions

Question

5c²- 24cd-5d²

Answer 1

Let me try to present that in a logical and reproducible format.

 

In order to do that I am going to change your c and d into x and y. So your equation is now:

 

5x² – 24xy – 5y²

 

Factoring means that we are going to try to rewrite this expression as...

 

(ax+by)(cx+dy), where a, b, c, and d are whole numbers.

 

If you distribute (FOIL) the terms above, you end up with...

 

acx² + adxy + bcxy + bdy²

 

This can be rewritten as...

 

acx² + (ad+bc)xy + bdy²

 

Comparing this generic expression to the one you have:

 

  5x² –      24xy     – 5y²

acx² + (ad+bc)xy + bdy²

 

You can see that:

 

ac = 5

bd = -5

ad+bc = -24

 

At this point it is easier to simply start to pick numbers for the "ac" pair and the "bd" pair and see if ad+bc comes out correctly as -24.

 

ac           bd           ad+bc

1;5        -1;5         5–5 = 0          <= this does not work

1;5        1;-5         -5+5 = 0        <= this does not work

5;1        -1;5         25–1 = 24      <= this does not work, but we are getting closer... only the sign is wrong.

5;1        1;-5        -25+1 = -24    <= this time it works... so now we know that:

 

a=5; c=1; b=1; d=-5

 

Plug that into the standard factoring system:

 

(ax+by)(cx+dy)

(5x+1y)(1x-5y)

 

To get back to your original variables... replace x with c and y with d and you are done.

 

 

 

Answer 2

Since the coefficient of d² is negative, the coefficient of c in the one of the factors is going to be positive and the coefficient of d in the other factor will be negative.  So it's going to look something like this:

 

(_c + _d)(_c - _d)

 

And because the coefficients of both c² and d² are 5, which is a prime number, that means that the coefficient of c in the one of the factors is going to be ±1 and the coefficient of c in the other factor will be ±5 -- and the same is true for d.

 

If you use both of the conclusions from above, you find that the only possibilities are:

 

(5c + d)(c - 5d)

(5c - d)(c + 5d)

(5c + 5d)(c - d)

(5c - 5d)(c + d)

 

When you multiply each of the pairs of factors, you find that the one that works is (5c + d)(c - 5d).