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 of 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#

‌

What is the degree of the polynomial #f(x)=-5 x^2-3 x-3#?

#2#

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 #2#.

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