Quickest way to factorize $\frac{w^2 + 5kw + 4k^2}{w^2+kw}$

Question

I would say 90% of polynomials in my textbook are factorable

e.g.

$$\frac{w^2 + 5kw + 4k^2}{w^2+kw}$$

This gives

$$\frac{(w+k)(w+4k)}{(w+k)w}$$ $$\frac{w+4k}{w}$$

This took me far too long to find.

I'd love to hear your approaches for quickly solving this type of problem.

Answer 1

I'd start off by breaking it into 2 parts:

$$1 + \frac{4k^2 + 4kw}{w^2 + kw}$$

Then I'd factorise the fraction:

$$1 + \frac {4k(k+w)}{w(k+w)}$$

The K+W cancels out and we're left with:

$$1 + \frac {4k}{w} = \frac {w + 4k}{w}$$

Answer 2

I assume you're fine factoring the denominator, so I'll focus on the numerator.

Here you can just treat $k$ like a fixed number, "pretending" you're just factoring a normal quadratic in $w$. Then your expression looks like $w^2 + \boxed{5k}w + \boxed{4k^2}$.

In usual quadratic-factoring fashion, that means you want a pair of numbers that multiply to $4k^2$ and add to $5k$.

• To add two things and get a multiple of $k$, probably each of those things must be a multiple of $k$, so I'd look for a pair of numbers that multiply to $4$ and add up to $5$, then you'll just stick $k$'s after them. This pair of number is of course $1$ and $4$, and if we check, $k \cdot 4k = 4k^2 \checkmark$ and $k + 4k = 5k \checkmark$.

So still in usual quadratic fashion, you've found two things that multiply to $4k^2$ and add up to $5k$, so you know it's going to factor as $(w + \boxed{\rm first\ thing})(w + \boxed{\rm second\ thing}) = (w + k)(w + 4k)$.