Polynomials

Functions: Higher degree polynomials

Polynomials

Poly‍nomials

A polynomial is a function of the form

$f(x)=a_nx^n+a_{n-1}x^{n-1}+ \ldots +a_2x^2+a_1x+a_0$

where $a_1$ , $a_2$ , $\ldots$ , $a_n$ are numbers $a_n \ne 0$ and $n$ is a positive integer.

We call $n$ the degree of the polynomial.

The numbers $a_1$ , $a_2$ , $\ldots$ , $a_{n-1}$ , $a_n$ are called the coefficients of the polynomial and $a_n$ is called the leading coefficient.

Examples

$\begin{array}{rcl}f(x)&=& 2x^2+3 \\ \\ g(x)&=&4x^5+3x^2-4x+6 \\ \\ h(x)&=&-\frac{1}{2}x^6+3x^4 \\ \\ k(x)&=&5\end{array}$

Examples degree

The degree of $f(x)= 2x^2+3$ is equal to $2$ . The leading coefficient is equal to $2$ .

The degree of $g(x)=4x^5+3x^2-4x+6$ is equal to $5$ . The leading coefficient is equal to $4$ .

The degree of $h(x)=-\frac{1}{2}x^6+3x^4$ is equal to $6$ . The leading coefficient is equal to $-\frac{1}{2}$ .

The degree of $k(x)=5$ is equal to $0$ . The leading coefficient is equal to $5$ .

Linear and quadratic

Previously we have seen linear and quadratic functions. These are both polynomials.

A linear function is a polynomial of degree $1$ . We can, therefore, also call it a first-degree function.

A quadratic function is a polynomial of degree $2$ . We can, therefore, also call it a second-degree function.

Brackets

With quadratic functions, we saw that we can recognize $f(x)=\left(x-2\right) \left(x+3\right)$ as a quadratic function by expanding the brackets.

Also, with polynomials, we expand the brackets to recognize that a function is a polynomial. In this way, it becomes obvious right away which degree the polynomial is.

Example

Take a look at the function $f(x)=x^2\left(x-3\right)\left(x+5\right)$ . We expand the brackets in this function.

$\begin{array}{rcl}f(x)&=&x^2\left(x-3\right)\left(x+5\right) \\ &&\phantom{xxx}\blue{\text{the given function}} \\ &=&x^2\left(x^2 +2x-15\right) \\ &&\phantom{xxx}\blue{\text{expanded double brackets}} \\ &=&x^4+2x^3-15x^2 \\ &&\phantom{xxx}\blue{\text{expanded single brackets}} \end{array}$ Hence, the function $f(x)=x^2\left(x-3\right)\left(x+5\right)$ is a polynomial of degree $4$ .

Graph

The graph of a polynomial of degree $n$ has no more than $n$ zeros and no more than $n-1$ vertices. It can also have less.

Example

$f(x)=x^4-3x^2+2$

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What is the degree of the polynomial $f(x)=6 x+1$ ?

$1$

A polynomial is of the form $f(x)=a_nx^n+a_{n-1}x^{n-1}+ \ldots +a_2x^2+a_1x+a_0$ . In which $a_1$ , $a_2$ , $\ldots$ , $a_n$ are number and $a_n \ne 0$ and $n$ is the degree of the polynomial.
In this case, the degree is equal to $1$ .

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