I'm not sure what exactly your approach would be. Originally, I just posted my approach:

I am sure that his approach is

because that's also what I would do.]]>Clever and compact.

]]>I couldn't agree more that math should be neat, clean and simple enough for everyone to grasp.

That's been my passion and my focus most of my career --- viewing math as a language and

trying to see if the symbolism, definitions, algorithms, etc. can be simplified and improved. It

really needs close scrutiny since it evolved over many centuries by folks that had no chance to

communicate with each other to try to make it really consistent, coordinated, correct, concise, and

any other word we can think of that starts with a "c".

It is nifty. Neat and clean and simple enough for everyone to grasp. That is what math should be like.

]]>I've never seen the generalized "sum of roots" formula you gave in post#6. That's really nifty!

Thanks for sharing it.

You can use the theory of equations to try another way. It says there is a relationship between the roots and the coefficients.

The sum of the roots of a nth degree polynomial

are equal to

So you have this equation to solve

solving for r3 you get r3 = -1 which is the third root.

]]>Also you could call the quadratic coefficients a, b and c and do a bit of algebra to get them, but it's still pretty much the same.

In short, I think you have the optimum method already.

Bob

]]>That approach looks like the most straight forward to me.

For this particular problem

you might notice that

which means you know (x+1) is factor straight away by the factor theorem.

In general, dividing by known factors is the way.

Bob

ps. For typical exam questions, they cannot choose factors that would take a long time to find, so I always do a quick mental check for x = +/-1, +/-2, +/-3. If I haven't found a factor by then I do another quicker question first.

]]>If 2 is a zero of the polynomial, then (x - 2) is a factor.

(By polynomial long-division)

Therefore, the other zero is -1

But having re-read your first post, I think that might be what you would have done anyway. I'm not sure that I know any more efficient method. But I suggest you just divide once and then factorise, that - at least - might make things a little faster?

]]>and the polynomial is

What is the third zero?

My approach to such kinds of problem is to divide the polynomial by the two given factors to obtain the other factor and then the zero.

*Can Someone suggest a quicker and more efficient method?*