# How do you factor # rs - rt - ks - kt#?

##### 1 Answer

#### Answer:

#### Explanation:

This is an interesting question in that it looks like a trick question or a typo.

For example,

#rs-rt-ks+kt = (r-k)(s-t)#

#rs-rt+ks-kt = (r+k)(s-t)#

but

#rs-rt-ks-kt#

cannot be factored further.

**Sketch of a proof**

Since all of the terms are of degree

Since there are no terms in

Since there are no terms in

Hence up to scalar factors, the factorisation must be expressible in the form:

#rs-rt-ks-kt = (r+ak)(s+bt) = rs+brt+aks+abkt#

for some constants

Equating coefficients we find:

#{ (b = -1), (a = -1), (ab = -1) :}#

which is inconsistent, since

So there is no such factorisation.

**Random Advanced Footnote**

It is actually possible to factor

#rs-rt-ks-kt = rs+rt+ks+kt = (r+k)(s+t)#