Today, I was working on some limit practice problems and came across two that I had to factor.
The first limit had this polynomial in the denominator: $$x^2+2x-15$$
which I factored down to: $$(x-3)(x+5)$$
The second limit had this polynomial in the numerator: $$2z^2-17z+8$$
which I factored down to: $$(2z-1)(z-8)$$
As I was looking over these problems, I realized I don't know why polynomials factor down like this. I was just taught what to do when I come across each type. When I factor them down the answers "make sense", but I just can't see a reason why they do as I look at it from my current perspective. Could someone shed some light on this? Are there proofs for things like this?
Best Answer
Perhaps you're noticing the streetlight effect:
What's the point of factoring a polynomial? It's to undo polynomial multiplication. Our streetlight is our knowledge of polynomial multiplication. How do we get a quadratic that we can factor "nicely" (ie over the integers)? Like this: $$ (ax + b)(cx + d) = acx^2 + (ad + bc)x + bd $$ So if we have a quadratic that we can factor, we had better be able to find four numbers $a,b,c,d$ such that $ac$ is the coefficient of $x^2$, $bd$ is the constant term, and $ad + bc$ is the coefficient of $x$.
That's the ultimate root of the techniques you learned for factoring different types of quadratics over the integers. There are many quadratics that aren't under the streetlamp, so to speak, such as $x^2 - 2$ and $x^2 + 1$; in broadening the circle of light, we meet the real numbers, the quadratic formula, and the complex numbers.