Solution of equation $(1 - c)[(2 - c) ^ 2 - 9] = 0$

Question

I was solving a characteristic equation for matrix eigenvalue problem. I had this equation, where $c$ is eigenvalue.

$$(1 - c)[(2 - c) ^ 2 - 9] = 0$$

In the book they equated both the terms to zero, like this

$$(1 - c) = 0$$

$$[(2 - c) ^ 2 - 9] = 0$$

And find out $c = 1, 5, -1$.

I am unable to understand how they can equate both terms to zero like this?

Answer 1

When you have a product of terms that is $0$, like $ab=0$, you know that at least one of $a,b$ is $0$. They break it apart into two equations but you only know that at least one is true. You solve each equation and the solution set of the original is the union of the solution sets of the separated equations. Here the first gives $c=1$ and the second gives $c \in \{5,-1\}$. The solution to the original is then $c \in \{1,5,-1\}$

Answer 2

We call this the zero product property.

In high school algebra, you're tasked with finding solutions to quadratic equations $x^2 + bx + c = 0.$ One way we first introduce this is by factoring, say by finding $p, q$ such that $(x-p)(x-q) = x^2 + bx + c.$ By equating coefficients, you'll see that $pq = c$ and $p+q = b.$ So why does this help?

In the real number system, the number $0$ plays a nice role as a sort of "annihilator." Multiplication by zero produces, well, zero. So what can we say if we multiply two numbers and get zero? That is, if $uv = 0$ then what can be said about $u$ and $v?$ Certainly they both can't be different from zero at the same time, but we need one of the two (maybe even both!) to be zero. This gives the general idea: if you multiply a bunch of numbers and get $0$ as a product, at least one of those numbers must be $0.$

So how does this help in your question? Well, if $(1 - c)[(2 - c) ^ 2 - 9] = 0$, then by the above reasoning, we must have either $1 - c = 0$ or $(2-c)^2 - 9 = 0,$ and both are easy to solve! Why do we have this? Two expressions multiply together and give zero. So at least one (possibly both) must be equal to zero. Then we use this to get our values of $c.$

TL;DR: High school algebra with the zero product property.