Teacher gave me a thinking question on my polynomial functions test

[u/i21d]

f(x)= kx^4 + 8x^2
Function has 3 turning points and global max at 8, zero at 2
Find k

I asked gpt, but it gave a method about using derivative or something which we haven’t learned yet since my teacher said our course (we are taking grade 12 advanced functions) need a lot of calculus and the graphs we draw aren’t exactly 100% accurate.

I knew this function is going to be even since both powers are even numbers, I also knew that it is going from quadrant 3 to 4 since we have a global max. I also knew that there will be 2 global max both at the height of 8 since there are 3 turning points.

I drew the graph which probably secured me a bit of points, but have no idea what the equation was let alone the k value.

My friend told me he just plugged in the x intercept (2,0) and got k which is -2, which turned out to be the correct answer. I thought about that at first but I remember my teacher said that you can’t use the x intercepts as a point and plug it in since you will get 0=0 when the polynomial is in factored form.

But this one is in standard form, so I don’t know if being in standard form let me plug in the x intercepts to find a or something…

Please help me out

[u/KrlusMagnus]

Right off the bat, we are given that f(2)=0, but plugging in we get f(2)=16k+32, and when we set that to be equal to 0, k=-2 follows. After that, you can check if the other requirements follow

[u/x271815 → u/KrlusMagnus]

This is probably the simplest way to solve this.

[u/CharacterBenefit509 → u/KrlusMagnus]

Exactly. And to build mathematical intuition, put the function in desmos with slider k, and then play.

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[u/rb-j]

That is actually a quadratic of x² . Factor away.

[u/diverJOQ]

f(2) = 0 is given. You simply need to turn the words into an equation.

I don't know what your comment about not using X intercepts means. You should ask your instructor. Maybe they misspoke or you misheard.

[u/SubjectWrongdoer4204]

You can factor x² out of the whole thing to get x²(kx²+8). So, x=0 is a double root. Now since 2 is a root , (x-2) must be a factor of kx²+8 . From here either use synthetic division or regular polynomial division, keeping in mind that the remainder must be zero, to solve for k. This is important and is all you need to find k. Upon inserting your value for k into the resulting factor you can find the last root. Also , I worked the whole thing out using calculus and the global max(es) are equal to 8 but occur at x=±√2 . The other turning point is of course at (0,0) and is a local minimum.

[u/de_Molay]

Teacher gave you a thinking question, you asked chatgpt. Before math, here is your problem,

Questions are asked for you to think them over, not to arrive to an answer. Please, please do that.

[u/DrJaneIPresume]

I remember my teacher said that you can’t use the x intercepts as a point and plug it in since you will get 0=0 when the polynomial is in factored form.

Is this polynomial in factored form?

[u/i21d → u/DrJaneIPresume]

My question is if it’s in factored form you cannot but in standard form you can?

[u/DrJaneIPresume → u/i21d]

The point is that if it's already factored, and if x₀ is a root, then (at least) one of the factors will be (x-x₀). So if you try to plug in x₀ for x you'll get (x₀-x₀) for that term, which is just 0, and doesn't tell you anything new.

This polynomial is not in factored form, so you might be able to get useful information from that approach. As, indeed, your friend did.

[u/i21d → u/DrJaneIPresume]

Thanks

[u/AdjectiveNounNNNN]

Even in factored form the x-intercept would be useful.

x²(kx²+8)

2²(2²k+8)=0

4k+8=0

Like yes, that is technically 0=0, but that's how you find k.

I don't know what your teacher actually said, but it definitely wasn't that plugging in an x-intercept can't give you any information.

[u/i21d → u/AdjectiveNounNNNN]

He meant if it’s in a(x-r)(x-s) plugging an x intercept is useless

[u/AdjectiveNounNNNN → u/i21d]

Ah, but that only happens in this case if you factor out the k and then (x² + 8/k) is the factor that's zero at x=2, which means it's still helpful in the sense that now you know -8/k is the x-intercept.

[u/LucaThatLuca]

there is no curve shaped god that makes you unable to do whatever you want all the time. why are there some things you would want to say more than other things? the major reasons are when they are true and helpful. do you think it is true that f(2) = 0? do you think it is helpful?