# Show that `25m^2-40mn+16n^2` is non-negative for all values of m and n.

*print*Print*list*Cite

### 2 Answers

To prove that `25m^2-40mn +16n^2` is non-negative for all values of m and n, set the polynomial greater than and equal to zero.

`25m^2-40mn +16n^2gt=0`

Then, factor.

`(5m-4n)(5m-4n)gt=0`

`(5m-4n)^2gt=0`

Then, consider this property of exponent which is:

`x^ngt=0 ` when n is an even number.

Since the exponent of the factor `5m-4n` is 2 which is an evennumber, then the inequality equation `(5m - 4n)^2gt=0` is always true.

**Hence, `25m^2-40mn +16n^2` is non-negative for any values of m and n.**

`25mn^2-40mn+16n^2=(5m-4n)^2>=0`