To tackle this question, we'll want to see the number sense behind factoring by inspection. In the quadratic construction 

we're looking for two numbers that multiply to give us our constant, "c," and sum to give us the coefficient "b."

In this case, those two numbers are 9 and -4, meaning we can reconstruct the function to read

since we're looking for our x intercepts or "zeros," we want the case where our output or "y" = 0, so we can set the righthand side of the function equal to zero as such

in order for this equation to hold true, either of our parentheticals will need to equal zero, meaning that if

or

then x could be equal to -9 or 4. So, I and III are zeros of the function. 

*Note - if given this question on a calculator-friendly section, we could graph and identify zeros as well*

The important first step of creating a quadratic equation from its zeros is knowing what a zero really is. A zero of a quadratic (or polynomial) is an x-coordinate at which the y-coordinate is equal to 0.  

We find that algebraically by factoring quadratics into the form , and then setting  equal to  and , because in each of those cases and entire parenthetical term would equal 0, and anything times 0 equals 0.

Here the question gives you a head start: we know that the numbers 4 and 5 can go in the  and  spots, because if so we'll have found our zeros. So we can set up the equation:

This satisfies the requirements of zeros, but now we need to expand this equation using FOIL to turn it into a proper quadratic. That means that our quadratic is:

And when we combine like terms it's:

The zeros of a function are points at which the function crosses the x-axis, or perhaps more simply points at which the function is equal to 0. So to solve for those, set the function equal to zero and then solve it like you would a quadratic. Here that gives you:

You can then factor the common  term:

And then factor the quadratic within parentheses. Note that this quadratic is one of the common perfect square quadratics:

 or  

The solutions to this equation, then, are  and . Note that the question asks you for how many distinct zeros the function has, so you cannot count  twice. The answer, then, is 2.

A function with zeros at 2 and 7 would factor to  , where it is important to recognize that  is the coefficient. So while you might be looking to simply expand   to , note that none of the options with a simple  term (and not ) directly equal that simple quadratic when set to 0.

However, if you multiply  by 2, you get . 

When you're graphing a function, the zeros of the function are the points at which the function crosses the x-axis. What that really means is that the value of the function is zero at those points, so to solve for those zeros algebraically you can just set the function equal to zero and solve. Here that would mean:

So factor the common  term to get:

And then factor the quadratic within parentheses:

You can then see that there are three values of  that would make this equation true:  and . The answer is therefore 3.

The zeros of a function are the x-values at which the function is equal to zero. So to solve for the zeros, set the function equal to zero. That would give you here:

Then you can factor the common  to get:

And then factor like you would a quadratic:

Or, more succinctly formatted:

This means that the zeros are at  and . Now, importantly, look at what the question asks for. It wants the sum (add the zeros) of all UNIQUE zeros, meaning you should not count  twice. The sum then is .

The zeros of a function are points at which the function is equal to zero. Here you're told that  does not equal zero for any real number values of  and you're asked to determine what that means for . Note that since  cannot be negative,  would have to be negative in order for the  term to reduce the other term,  to zero. Since you know that this function never equals zero, you can conclude that  is not negative, which means that it is greater than or equal to zero.

The zeros of a function are the x-values at which the function itself is equal to zero, so you will generally want to solve these problems by setting the function equal to zero and then factoring like a quadratic. Here that means you would start with:

And then factor the common  term:

You now know that one of the solutions is , as that would mean that the entire parenthetical term would be multiplied by zero. But you still need to work within the parentheses to factor that quadratic. You should see some helpful factoring clues: when you factor into two parentheticals, the numeric terms have to multiply to -1, meaning that you will likely have one be 1 and the other -1. And the first terms need to multiply to  meaning that your first terms will be  and . So you can set up your parentheses as:

, where one  will have a + and one will have a - sign.If you play with the options to see what will work to give you the  middle term in , you'll see that the proper factorization is:

This means that the solutions for  are  and .  The sum of these solutions, then, is .

The zeros of a function are essentially points (or at least the x-values of the points) at which the function is equal to zero. So to solve for the zeros of a function, first set that function itself equal to zero. Here that would mean:

Then factor like you would a quadratic; since you have it set to zero, if any multiplicative term equals zero then the "equals zero" will hold for the whole equation. First you can factor the common  term:

And then you can factor the quadratic within:

This then means that the zeros for this function are at  and , meaning that this function has three distinct zeros.

The zeros of a function are the x-coordinates at the points where the function (the y-coordinate, if you're graphing it) equals zero. So to solve for the zeros, set the function equal to zero:

Here you can then factor the common  term, yielding:

Now you can factor the quadratic within the parentheses. This gives you:

The solutions to this equation are at the points  and , demonstrating that there are 3 unique zeros to this function.