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What is the degree of the polynomial ​?

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What is the degree of the polynomial \((3x^2 +11)^{12}\)?

May 4, 2024

#1
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This is actually quite an easy question!

At first, it might look like we have to distribute out and actually go through and raise 3x^2 + 11 to the power of 12, which is insanely time consuming and honestly, just not worth it.

However, there is a simple trick we could use to solve this problem. Notice that the problem is asking for the degree of polynomial, which is the highest power in this polynomial.

This means we wouldn't have to actually find the entire polynomial.

Now, let's note something. We see the orinal polynomial has degree 2, since it's leading term has a power of 2. This means that not matter what, it is not possible to get a higher degree than whatever this orginal degree is raised to the power of. What I mean is, when we raise 3x^2 to the power of 12, the other terms in the polynomial will not have a higher power than that!

So the highest degree is just (x^2)^12, which is just x^24. So the new polynomial has degree 24!

I hope you understood my explenation!

Thanks!

May 4, 2024