Degree of a Polynomial

[markraz]

hey
a polynomial such as 4x^3+3x^2-1 has a degree of 3, correct?
which means it will hit 0 on the Y axis 3 times since it has 3 roots.

So my question is: if you have a rational with polynomials in the the numerator and denominator
how do you calculate the degree?

for example: what would the degree be here? how many times will it land on the Y axis?
(4x^3+3x^2-1)/(4x^4+3x^2-1)

 

[Quaid]

a polynomial such as 4x^3+3x^2-1 has a degree of 3, correct?

Hi Mark. You're correct, here.

which means it will hit 0 on the Y axis 3 times since it has 3 roots.

You meant to type X axis, above, yes?

Only third-degree polynomials with three distinct, Real roots have three x-intercepts.

The polynomial 4x^3+3x^2-1 does not fall into this category. Two of its three roots are non-Real; in cases like this, there is only one x-intercept.

Other possibilities exist, for polynomials of degree three. All three roots could be the same Real number (see "multiplicity"); in this case also, there would be only one x-intercept. An example of such a polynomial is x^3+12x^2+48x+64.

Another possibility would be two Real roots, one of which is repeated (multiplicity 2); in this case, there would be two x-intercepts. An example of such a polynomial is x^3+7x^2+8x-16.

The lesson is: don't assume that nth degree means n x-intercepts.

So my question is: if you have a rational with polynomials in the the numerator and denominator how do you calculate the degree?

Polynomials have degree; ratios of polynomials do not.

Calculating the x-intercept(s) of y=(4x^3+3x^2-1)/(4x^4+3x^2-1) is beyond beginning algebra. Best you can do is zoom in on a graph, to get decimal approximation(s).

If you can find a way to simplify a ratio of polynomials, or the roots of the numerator are easy, then maybe exact x-intercepts are within reach. Or, maybe there are no x-intercepts, as in y=x^3/x^4.

Cheers :cool: