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Understanding the general facts about polynomials of degree n.

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You are given the equation {poly}.

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$\\phantom{a}$

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The equation {poly} can be described as a

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An equation of the form $y=c_nx^n+c_{n-1}x^{n-1}+\\ldots+c_1x+c_0$ is called an $n$th degree polynomial.

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$0$th degree polynomial

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polynomial of degree $1$

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polynomial of degree $2$

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hyperbola

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circle

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polynomial of degree $\\var{n}$

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As we move to the far left of the graph of {poly}, the graph

What happens to the graph as you go far to the left or right is called the long term behaviour of a graph.

The leading term (the term that includes the highest power) determines the long term behaviour of a polynomial. For our polynomial this is $\\simplify{{c[n]}x^{n}}$.

As we go far to the left of the graph $x$ is negative, and so $\\simplify{{c[n]}x^{n}}$ is negative. That is, the graph goes downwards. is positive. That is, the graph goes upwards.

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goes upwards.

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goes downwards.

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As we move to the far right of the graph of {poly}, the graph

What happens to the graph as you go far to the left or right is called the long term behaviour of a graph.

The leading term (the term that includes the highest power) determines the long term behaviour of a polynomial. For our polynomial this is $\\simplify{{c[n]}x^{n}}$.

As we go far to the right of the graph $x$ is negative, and so $\\simplify{{c[n]}x^{n}}$ is negative. That is, the graph goes downwards. is positive. That is, the graph goes upwards.

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goes upwards.

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goes downwards.

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The $y$-intercept of the graph of {poly} is $y=$[[0]].

The $y$-intercept is the value of $y$ when $x=0$, that is, the value of $y$ where the graph hits the $y$-axis. To find it, substitute $x=0$ into our equation {poly}. Doing so shows that $y=\\var{c[0]}$ is the $y$-intercept.

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Given any polynomial of degree $\\var{n}$, the maximum number of $x$-intercepts in its graph is [[0]] .

The $x$-intercept is the value of $x$ when $y=0$, that is, the value of $x$ where the graph hits the $x$-axis. To find it, substitute $y=0$ into our equation:

{poly0}

The Fundamental Theorem of Algebra says there are exactly $\\var{n}$ (complex) solutions to this equation (including multiplicity). The $x$-intercepts for our polynomial are real solutions to the above equation and therefore there are at most $\\var{n}$ $x$-intercepts.

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Given any polynomial of degree $\\var{n}$, the maximum number of possible 'bends' or 'turns' in the graph is [[0]].

A degree $n$ polynomial has at most $n-1$ bends in its graph.

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