Fx equation 54/27/2023 ![]() So the possible number of real roots, you could have 7 real roots, 5 real roots, 3 real roots or 1 real root for this 7th degree polynomial. And then finally, we could consider having 0 real and 7 non-real complex and that's not possible because these are always going toĬome in pairs, so you're always going to have an even number here. And then you could go toġ real and 6 non-real. And then we can go to 2 and 5, once again this is an odd number, these come in pairs, ![]() Then if we go to 3 and 4, this is absolutely possible. To have an even number of non-real complex roots. This is not possible because I have an odd number here. So for example,this is possible and I could just keep going. Essentially you can haveĪn odd number of real roots up to and including 7. Of course is possible because now you have a pair here. Have 2 non-real complex, adding up to 7, and that ![]() Now what about having 5 real roots? That means that you would Pairs, conjugate pairs, so you're always going to have an even number of non-real complex roots. This because the non-real complex roots come in Now could you have 6 real roots, in which case that would imply that you have 1 non-real root. So you could have 7 real roots, and then you would have no non-real roots, so this is absolutely possible. That you're talking about complex numbers that are not real. The reason I'm not just saying complex is because real numbers are a subset of complex numbers, but this is being clear So real roots and then non-real, complex. What that would imply about the non-real complex roots. To have 6 real roots? Is 6 real roots a possibility? Is this a possibility? Well, let's think about Or add 2 to both sides, or add 9, or subtract 3.5, or multiply by 617.8, etc. So you could add 1 to both sides, and now its written a new way. If you graphed this out, it could potentially You can rewrite equations by performing algebraic manipulations, making sure you always do the same operation on both sides of the '' sign. So we're definitely not going to have 8 or 9 or 10 real roots, at most we're going to have 7 real roots, so possible number of real roots, so possible - let me write this down - possible number of real roots. Going to have 7 roots some of which, could be actually real. Number of real roots? For example, could you have 9 real roots? And so I encourage you to pause this video and think about, what are all the possible number of real roots? So I'm assuming you've given a go at it, so the Fundamental Theorem of Algebra tells us that we are definitely It's clearly a 7th degree polynomial, and what I want to do is think about, what are the possible number of real roots for this polynomial right over here. Use horizontal line test to determine whether the function f ( x) x 2 is one-to-one. Our math solver supports basic math, pre-algebra, algebra, trigonometry. ( f) ( x) 1 2 ( x 2) Thus, the given function f ( x) x 2 is observed to be differentiable for x > 2. Solve your math problems using our free math solver with step-by-step solutions. (a) Obtain the derivative of the inverse function, f ( x) x 2. We have a function p(x)ĭefined by this polynomial. Step 3: Use the Horizontal line test by plotting the graph.
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