How Do I Find Asymptotes and Construct Polynomials with Given Zeros?

[HawKMX2004]

Ok, I have a final in Pre-Calc comming up, and I am still a bit confused on finding asymtotes (vertical, horizontal, and slant) could someone help me with equations i can use to find the asymtotes or how i do? I am just really confused.

Also, I am having trouble with finding a fourth degree polynomial that has a set of given zeros. How might i go about solving one of those? I am very confused, please help me

 

[Nylex]

I've answered your question in the K12 subforum. Don't post threads in more than one place, cos the mods will lock them.

 

[M98Ranger]

Asymptotes and polynomials can be tricky concepts to grasp, so don't worry if you're feeling confused. Let's start with asymptotes. An asymptote is a line that a graph approaches but never touches. There are three types of asymptotes: vertical, horizontal, and slant.

To find the vertical asymptote of a rational function (a function with a polynomial in the numerator and denominator), set the denominator equal to zero and solve for x. The resulting value of x is the equation of the vertical asymptote.

To find the horizontal asymptote of a rational function, compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

To find the slant asymptote of a rational function, use long division to divide the numerator by the denominator. The resulting quotient is the equation of the slant asymptote.

Now, onto finding a fourth degree polynomial with given zeros. If the zeros are real numbers, you can use the factor theorem to write the polynomial in factored form. For example, if the zeros are 2, -3, and 5, the polynomial would be (x-2)(x+3)(x-5). If the zeros are complex numbers, they will come in conjugate pairs. In this case, you can use the complex conjugate theorem to write the polynomial in factored form. For example, if the zeros are 2+3i and 2-3i, the polynomial would be (x-(2+3i))(x-(2-3i)).

Once you have the polynomial in factored form, you can expand it to get the final polynomial. Remember that the degree of a polynomial is equal to the sum of the exponents of its terms. So for a fourth degree polynomial, you will need four terms with exponents that add up to 4.

I hope this helps you better understand asymptotes and polynomials. Good luck on your final!