trouble with graphs of polynomials.

**Recyclop** · December 4th 2012, 05:12 PM

so im having trouble understanding the concept of assuming the graphs of polynomials. i wish i had exmaple questions but i dont, so i thought someone here could help.

things i dont understand:
- how to tell the least degree of a polynomial function.( i originally thought that it was by how many roots there were but doesnt seem to work for me.)
- how to tell whether the leading coefficient is positive or negative.
any help is appreciated. thanks

**topsquark** · December 5th 2012, 08:19 AM

Originally Posted by Recyclop

so im having trouble understanding the concept of assuming the graphs of polynomials. i wish i had exmaple questions but i dont, so i thought someone here could help.

things i dont understand:
- how to tell the least degree of a polynomial function.( i originally thought that it was by how many roots there were but doesnt seem to work for me.)
- how to tell whether the leading coefficient is positive or negative.
any help is appreciated. thanks

Consider the general polynomial $a_nx^n + a_{n - 1}x^{n - 1} + ~...~ + a_1x + a_0$

The least degree of the polynomial is the lowest power of x appearing in the polynomial. In this case 0. (Technically the last term of the polynomial can be written as $a_0 x^0$ . )

The leading coefficient is the coefficient of the highest power in the polynomial. In this case $a_n$ .

So if we have the polynomial $3x + 5x^3 - 8x^4$ what is the least degree of this function? What is the leading coefficient?

-Dan

**emakarov** · December 5th 2012, 09:29 AM

I am not sure there is a universally accepted concept of "the least degree" of a polynomial. The degree of a polynomial is the highest degree of its terms, where the degree of a term $x^n$ is $n$ . Thus, the degree of $a_nx^n + a_{n - 1}x^{n - 1} + ~...~ + a_1x + a_0$ is $n$ .

It's difficult to determine the degree from a graph when the axes are not graduated. For example, the graphs of $x^2$ and $x^4$ look similar. The number of real roots of a polynomial does not exceed its degree, but it can be strictly less than the degree.

For any polynomial $f(x)$ , $|f(x)|\to+\infty$ as $x\to\pm\infty$ . You can tell whether the degree of $f(x)$ is even or odd by observing the behavior at ±∞: if $f(x)$ tends to infinities of the same sign when x tends to +∞ and -∞, then the degree is even; if, for example, $f(x)\to+\infty$ as $x\to+\infty$ and $f(x)\to-\infty$ as $x\to-\infty$ , then the degree is odd.

The leading coefficient of $f(x)$ is positive iff $f(x)\to+\infty$ as $x\to+\infty$ , and it is negative iff $f(x)\to-\infty$ as $x\to+\infty$ .

**Recyclop** · December 5th 2012, 12:28 PM

@topsquark,
i understand how to find those things while seeing the polynomial infront of me the problem is interpreting that infromation from a graph.
for the polynomial you gave me the least degree would be 1, and the leading coefficient would be -8.

@emakarov i think im starting to understand now, thanks i will continue to use your post as a reference as i study for my finals

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