I was trying to complete the Full Quant Section #6 Quiz from Greg, and I found the following question (#8)

f(x) = x^8 - 7x^4 + 12

Quantity A:
The number of times f(x) intersects the x-axis in the xy-coordinate plane

Quantity B:
4

After seeing his video solution, I still don’t get it. Shouldn’t the function cross the X axis 8 times since the highest order in the variable is 8? Now, if I go to www.wolframalpha.com and plot this function, I clearly see that visually it crosses the X axis 4 times. What I want to know is what should I do to determine that it crosses the X axis 4 times? How do I handle similar problems in the future?

It’s true that x^8 = 8 will the most times this equation will touch the x axis but it will include both real + imaginary solutions and I think as GRE only care about real number the answer is 4 as it will have on 4 real solution. Maybe we can use a small disclaimer in the question that says real solution only for people who are new to GRE @Leaderboard .

Solve it the same way you solve a quadratic equation

For eg:

x^8-7x^4+12=0

Now, we need two numbers such that they added upto to -7 and their product is 12
Thus, two number will be -4,-3. Now, just square-root the highest degree polynomial in your equation which for our equation is x^8 thus, \sqrt{x^8}=x^4 hence, our equation becomes :

(x^4-4)(x^4-3) =0

Now, you can solve it! or use substitution like Greg did in the solution to arrive at the same!