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Question

What is an example of a polynomial you could add to h(x) to generate a polynomial with a degree of 5?

h(x) = -3x^6 + 2x^4 - 3x^2

Answer 1

The degree of a polynomial is just the highest power of x that is actually still there after everything is combined, so in order to end up with degree 5, two separate things have to happen at the same time.

The first one is that the -3x⁶ has to go away. As long as an x⁶ term is sitting in there, the degree is 6 no matter what we do with the rest of it, so we cancel it by adding its opposite, since -3x⁶ + 3x⁶ is 0.

But that by itself is not enough, and this is the part that is easy to miss. If we only added 3x⁶, we would be left with 2x⁴ - 3x², and that is degree 4, not 5. If you notice, h(x) does not have an x⁵ term anywhere in it, so if we want one in our final answer, we have to bring it in ourselves.

So we add both pieces, 3x⁶ to cancel and x⁵ to supply the degree we want:

(3x⁶ + x⁵) + (-3x⁶ + 2x⁴ - 3x²) = x⁵ + 2x⁴ - 3x²

and that is degree 5, which is what we were after.

So, one polynomial that works is 3x⁶ + x⁵. The coefficient on the x⁵ can be any number you like except 0, so something like 3x⁶ + 7x⁵ would work just as well, but the 3x⁶ part is not optional since that is the piece doing the cancelling.

Answer 2

Hello Jacqueline G.,

A polynomial of degree 5 would have a highest exponent of x equal to 5. So you have to cancel out the -3x⁶.

What would you add to -3x⁶ to cancel it out? Also, since there is not term with x ^ 5, you will also need to add a nonzero term with x ^ 5. (It could just be x ^ 5.) See if you can find the answer. You may submit your final answer to be checked by an expert.

Answer 3

Let take polynomial g(x) = 3x⁶ + x⁵; then h(x) + g(x) = - 3x⁶ + 2x⁴ - 3x² + 3x⁶ + x⁵ = x⁵ + 2x⁴ - 3x² and we get 5th degree polynomial

Answer 4

Add any polynomial that cancels the -3x⁶ term and adds a term with x⁵ in it. Example:

g(x) = 3x⁶ + x⁵

g(x) + h(x) = (3x⁶ - 3x⁶) + x⁵ + 2x⁴ - 3x² = x⁵ + 2x⁴ - 3x²

You are left with a degree 5 polynomial.