Question

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Answer 1

Answer: Choice A)  x^4+20x^2-100

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Explanation:

Choice B has the GCF x we can factor out like so

10x^4-5x^3+70x^2+3x = x(10x^3-5x^2+70x+3)

Showing that choice B is not prime. If a polynomial can be factored, then we consider it not prime. It's analogous to saying a number like 15 isn't prime because 15 = 3*5, ie 15 can be factored into something that doesn't involve 1 as a factor.

In contrast, we consider 7 prime because even though 7 = 1*7, there aren't any other ways to write this integer as a factorization if we don't involve 1.

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Choice C is a similar story. This time we can factor out 3

3x^2 + 18y = 3(x^2 + 6y)

So we can rule this out as well.

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Choice D is a bit tricky, but we can use the difference of cubes factoring rule

a^3 - b^3 = (a-b)(a^2+ab+b^2)

where in this case a = x and b = 3y^2

Note how b^3 = (3y^2)^3 = 3^3*(y^2)^3 = 27y^(2*3) = 27y^6

All of this means choice D can be factored and it's not prime either.

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We've ruled out choices B through D. The answer must be choice A.

If you let w = x^2, then w^2 = x^4

The polynomial w^2+20w-100 is prime because setting it equal to zero and solving for w leads to irrational solutions. I'm assuming your teacher wants you to factor over the rational numbers.

Because w^2+20w-100 can't be factored over the rational numbers, neither can x^4+20x^2-100. This confirms that choice A is prime.

Answer 2

A
This is like a base form of a prime polynomial
X^2+x+1