3.04 Introduction to polynomials

A polynomial is a mathematical expression with many terms ("poly" means "many" and "nomial" means "names" or "terms"). A polynomial can have any combination of operators (addition, subtraction, multiplication or division), constants, variables and integer exponents, but never division by a variable. This means that expressions with negative indices can never be polynomials because $x^{-a}$x−a is the same as $\frac{1}{x^a}$1xa​. 

Parts of a polynomial

Degree: The largest exponent (i.e. power) of a variable in a polynomial. e.g. In the polynomial $x^3+4x^2-9$x3+4x2−9, the highest power of $x$x is $3$3, so the degree in this polynomial is $3$3.

Leading coefficient: When a polynomial is written with its exponents in descending order, the leading coefficient is the number that is written before the first algebraic term. For example, in $5x-7$5x−7, the leading coefficient is $5$5. Sometimes you may need to use your knowledge of algebra to work out the leading coefficient. e.g. In $-x^5-2x^4+4$−x5−2x4+4, the leading coefficient is $-1$−1. When the leading coefficient is 1, the polynomial is called a monic polynomial.

Constant term: the term in a polynomial that has no variables (i.e. no algebraic terms). e.g. in the polynomial $4y^8+2xy-4x-\frac{2}{3}$4y8+2xy−4x−23​, the constant term is $-\frac{2}{3}$−23​.

Practice Questions

QUESTION 1

QUESTION 2

Sketching polynomials

If we start with the most basic form of a polynomial, $y=ax^n$y=axn, we can immediately see a pattern emerge for odd and even powers. Experiment with the following applet, where $a=1$a=1.

Created with Geogebra

Odd degree polynomials move in opposite directions at the extremities and even degree polynomials move in the same direction at the extremities! This is because the degree of a polynomial, $n$n, together with the leading coefficient, $a$a, dictate the overall shape and behaviour of the function at the extremities:

$n$n	$a>0$a>0	$a<0$a<0
Even
Odd

 

But what happens between the extremities? 

We know that a polynomial of degree $n$n can have up to $n$n $x$x-intercepts, with those of odd degree having at least one. 

A polynomial of degree $n$n can have up to $n-1$n−1 turning points, with those of even degree having at least one.

Details of key features can be found using Calculus and technology or from factorised forms of the function.  Let's focus on what we can learn from factorised forms of polynomials.

Factorised polynomials and multiple roots

Certain polynomials of degree $n$n can be factorised into $n$n linear factors over the real number field. For example the $4$4th degree polynomial $P\left(x\right)=2x^4-x^3-17x^2+16x+12$P(x)=2x4−x3−17x2+16x+12 can be re-expressed as $P\left(x\right)=\left(x+3\right)\left(2x+1\right)\left(x-2\right)^2$P(x)=(x+3)(2x+1)(x−2)2. Note that there are two distinct factors, the $\left(x+3\right)$(x+3) and $\left(2x+1\right)$(2x+1) and two equal factors of $\left(x-2\right)$(x−2) . We can immediately identify that there are roots at $x=-3,-\frac{1}{2}$x=−3,−12​ and $x=2$x=2.

Worked Examples

Example 1

The function $y=-\left(x-1\right)\left(x+2\right)^2$y=−(x−1)(x+2)2 has a root at $x=1$x=1 and a repeated root at $x=-2$x=−2. The $y$y - intercept is $-\left(-1\right)\left(2\right)^2=4$−(−1)(2)2=4. The function is of odd degree, so its ends move off in different directions. Note that for large positive values of $x$x, the curve becomes very negative (See graph $G3$G3 below).

Practice Questions

Question 3

Question 4

Question 5