End Behaviour of Polynomial Function

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We know End behaviour can refers to the behaviour of a graph, like it approaches either negative infinity or positive infinity. We know end behaviour of a polynomial function is determinate by degree of the function and leading coefficient.
If the degree of the polynomial is Even and leading coefficient is positive then
F(x) -> ∞, as x -> -∞ and also F(x) ->∞ as x -> +∞
If the degree of the polynomial is Even and leading coefficient is negative then
F(x) -> -∞, as x -> -∞ and also F(x) -> -∞ as x -> +∞
If the degree of the polynomial is Odd and leading coefficient is positive then
F(x) -> ∞, as x ->∞ and also F(x) ->-∞ as x -> -∞

Example 1: Find the end behaviour of the function  x<sup>4</sup>-4x<sup>3</sup>+3x+25

Solution: The given function is F(x) = x<sup>4</sup>-4x<sup>3</sup>+3x+25

The degree of this function is 4, its even number

We can see the leading coefficient,

That is positive.

So the end behaviour is

F(x) -> + ∞, as x -> -∞

F(x) -> + ∞, as x -> ∞

Example 2: Find the end behaviour of the function x<sup>3<sup>+x<sup>3<sup>+3x+2

Solution: The given function is F(x) = x<sup>3<sup>+x<sup>3<sup>+3x+2

The degree of this function is 3, its odd number

We can see the leading coefficient,

That is positive.

So the end behaviour is

If the degree of the polynomial is Odd and leading coefficient is positive then

F(x) -> ∞, as x ->∞ and also F(x) ->-∞ as x -> -∞

F(x) -> + ∞, as x -> -∞ and F(x) -> + ∞, as x -> ∞

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