Factor the expression completely.

Question

x^5 y^3 − x^4

I have seen multiple answers but do not understand how to get there.

Answer 1

factor out x^4

to get x^4 (xy^3 -1) as just 2 factors

or if you want

(x)(x)(x)(x)(xy³-1) as 5 factors

You could treat xy³-1 as (x^1/3y-1) as the difference of 2 cubes getting

more factors

(x^1/3y-1)(x^2/3y² + x^1/3y +1) but almost no one ever does that if you don't get integer powers by factoring

best answer is

x⁴(xy³-1) You get that by just dividing x⁴ into x⁵y³ - x⁴

You can think of it as multiplying by 1 = x⁴/x⁴ times x⁵y³ - x⁴ to get x⁴ (x⁵y³-x⁴)/x⁴ = x⁴(x⁵y³/x⁴ - x⁴/x⁴)

=x⁴(xy³-1)

You don't usually see anything further, but you could, if you really wanted to, also rewrite that breaking down all the powers into factors as

xxxx(xyyy-1)

or (x)(x)(x)(x)[x(y)(y)(y) - 1]

Answer 2

(x, x, x, x, x)(y, y, y)-(x, x, x, x)