Answer :
Answer with explanation:
The general polynomial function is given by:
[tex]P(x)=a_nx^n+a_{n-1}x^{n-1}+.....+a_1x+a_0[/tex]
where [tex]a_n\neq 0[/tex] and it is the leading coefficient.
Also, if n is even then it is a even degree polynomial.
and if n is odd then it is a odd degree polynomial.
The end behavior of a polynomial is the behavior when x tends to infinity from both the sides.
i.e. when x → -∞ and when x → ∞
Now,
If n is even:
1)
[tex]a_n\ \text{is positive}[/tex]
then if x → -∞ then P(x) → ∞
and if x → ∞ then P(x) → ∞
2)
[tex]a_n\ \text{is negative}[/tex]
then if x → -∞ then P(x) → -∞
and if x → ∞ then P(x) → -∞
If n is odd:
1)
[tex]a_n\ \text{is positive}[/tex]
then if x → -∞ then P(x) → -∞
and if x → ∞ then P(x) → ∞
2)
[tex]a_n\ \text{is negative}[/tex]
then if x → -∞ then P(x) → ∞
and if x → ∞ then P(x) → -∞
To describe the end behavior of polynomial graphs with odd and even degrees, generalised polynomial function can be used. When x tends to infinity from both the sides, shows the end behavior of the polynomial.
Generalised polynomial function can be used to describe the end behavior of polynomial graphs with odd and even degrees.
The generalised polynomial function is given by:
[tex]\rm P(x) = a_n x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+........+a_1x+a_0[/tex] --- (1)
When, n is even then equation (1) becomes even degree polynomial and when n is odd then equaation (1) becomes odd degree polynomial.
- Now, if n is even and [tex]a_n[/tex] is positive then if [tex]x \to -\infty[/tex] then [tex]\rm P(x)\to \infty[/tex].
- Now, if n is even and [tex]a_n[/tex] is negative then if [tex]x \to -\infty[/tex] then [tex]\rm P(x)\to -\infty[/tex].
- Now, if n is even and [tex]a_n[/tex] is positive then if [tex]x \to \infty[/tex] then [tex]\rm P(x)\to \infty[/tex].
- Now, if n is even and [tex]a_n[/tex] is negative then if [tex]x \to \infty[/tex] then [tex]\rm P(x)\to - \infty[/tex].
- Now, if n is odd and [tex]a_n[/tex] is positive then if [tex]x \to -\infty[/tex] then [tex]\rm P(x)\to -\infty[/tex].
- Now, if n is odd and [tex]a_n[/tex] is negative then if [tex]x \to -\infty[/tex] then [tex]\rm P(x)\to \infty[/tex].
- Now, if n is odd and [tex]a_n[/tex] is positive then if [tex]x \to \infty[/tex] then [tex]\rm P(x)\to \infty[/tex].
- Now, if n is odd and [tex]a_n[/tex] is negative then if [tex]x \to \infty[/tex] then [tex]\rm P(x)\to - \infty[/tex].
For more information, refer the link given below:
https://brainly.com/question/13053190