Please factor with two variables each polynomial: 12m^2n^2-8mn+1 and factor completely: w^3-3w^2-18w Thank you.

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llltkl | College Teacher | (Level 3) Valedictorian

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1. `12m^2n^2-8mn+1`

This is a polynomial with two variables where the powers of the variables are in a string. Split -8 as a sum of two terms so that their product is `12*1` , to obtain the given polynomial with two variables as:

`=12m^2n^2-8mn+1`

`=12m^2n^2-6mn-2mn+1`

Take 6mn as the common factor out from the first two terms and -1, from the next two terms, to obtain the given polynomial with two variables as:

`6mn(2mn-1) -1(2mn-1)`

=(2mn-1)(6mn-1)

2. `w^3-3w^2-18w`

Here too, the polynomials are in a string. Split the middle term, `-3` as a sum of two terms such that their product is `1*-18` , i.e. `-18` , such that the given polynomial becomes:

`= w^3-6w^2+3w^2-18w`

Take w^2 as the common factor out from the first two terms and 3w, from the next two terms, such that the given polynomial takes the form:

`=w^2(w-6)+3w(w-6)`

Now take (w-6) as the common factor out:

`=(w-6)(w^2+3w)`

Again take w as the common factor out, such that the given polynomial takes the factorized form:

=w(w-6)(w+3)

Sources:

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