factoring y^3+y^2?

Question

Factor the denominator y^3 + y^2 into linear factors. Factors of
y^3 + y^2 Enter your factors with their multiplicities as pairs [f,n].

having trouble understanding exactly what this is asking for

Answer 1

y^2(y+1)

Answer 2

1) A linear factor is a factor where the variable is of first degree
(y^2 is second degree, y^3 is third degree, and so on)

So
y^3 + y^2 = y*y*(y+1)

2) the term "multiplicities", when applied to a function, usually refers to the multiplicity of the roots: the values of the variable at which the function equals zero.

When a function is factored into linear factors, then each factor is of the form (y - a) where +a is a root.
Why? Because if +a is a root, then (y-a) = 0 and 0 times anything else will be zero.

So, we can rewrite our linear factors as:
(y-0)(y-0)(y+1)
and the roots are 0, 0, -1.

Note that I wrote 0 twice, because 0 is a root of multiplicity 2 (there are 2 linear factors for which 0 is a root).

This sounds weird for now, but the multiplicity of roots becomes important later when doing more advanced math.

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PS:

linear = degree one
square, quadric or quadratic = degree 2
cubic = degree 3
quartic = degree 4
quintic = degree 5
sextic = degree 6
and the list stops there
Ethymologically, the next one would be septic and that sounds too much like sceptic (as in sceptic tank).

Almost forgot:
constant = degree 0


In practice, most mathematicians will not use words beyond cubic. After that, we say: a polynomial of degree 4
for example:
x^4 -3*x^3 + 6*x^2 - 9*x + 18

Your problem started with a cubic (a polynomial of degree 3).

Answer 3

y³ + y² = y²(y + 1)
:>:

Answer 4

y^2(y + 1)

Answer 5

well if you factor out a y^2 you'll get

y^2(y+1)

I'm guessing that's what they are after... but its been a while since I've done factoring.

Answer 6

y(y+y) (y+y)???