Select the expressions that are polynomials. There may be more than one correct answer.

Question

14x^−9 − 10x^20 + 10x^1 − 7x^9

7x^18 − 13x^4 −14x^8 + 12/x^−9 + 8x^10

−8x^21 + 9x^3 −4x^27 +5x^16

−12x^12 −11x^11 −2x 4/7 + 3x^19

2x^24 + 6x^1.8 + 6x^14 + 1x^2

−5x^29 − 9x^6 + 15x + 7/x^18

Answer 1

Hi Nico,

The main thing to note with polynomials, which is apparently the highlight of this question, is that polynomials cannot possess variables with negative exponents and, furthermore, cannot contain fractional exponents.

For example:

2x³ - 1x² + x +1988 is a polynomial whereas 2x^-3 - 1x² + x +1988 is not a polynomial (note the negative sign in front of the 3 in the latter example).

This question is trying to trick you by putting negative exponents at the bottom of fractions which would make them polynomials indeed because they can be rewritten such that the exponent is positive.

Using an example form those you've provided above can help illustarte why this is tricky.

Look at 7x^18 − 13x^4 −14x^8 + 12/x^−9 + 8x^10

Focusing on the part I have made bold, we would not think this is a polynomial by virtue of the existence of x^-9 but since it is at the bottom of a fraction it can be rewritten to be at the top of the fraction in a positive form (x⁹)

In addition, polynomials cannot contain fractional exponents, so anything with an exponent like 2/3 or 4.8 is not considered a polynomial, as well.

So to answer the overall question we have:

14x^−9 − 10x^20 + 10x^1 − 7x^9 (not a polynomial)

7x^18 − 13x^4 −14x^8 + 12/x^−9 + 8x^10 (polynomial)

−8x^21 + 9x^3 −4x^27 +5x^16 (polynomial)

−12x^12 −11x^11 −2x^4/7 + 3x^19 (not a polynomial)

2x^24 + 6x^1.8 + 6x^14 + 1x^2 (not a polynomial)

−5x^29 − 9x^6 + 15x + 7/x^18 (polynomial)

I hope you found this helpful!