Polynomials are expressions created by the addition, subtraction or multiplication of variables.
Eg, x^3+ 2x^2 + 3 is a polynomial
x^1/2 + 2 is not a polynomial because it contains a rooted variable
x^-1 + 3x is not a polynomial because it contains a negative power
2^x + 5 is not a polynomial because it contains a variable as a exponent
^ is to the power of.
Polynomials can only have one variable and the powers of that variable must be positive integers.
Terminology
Roots – Also called zeroes, are the x intercepts or solutions of the polynomial.
Leading Term – The term in which the variable has the highest power
Leading Coefficient – The Coefficient of the leading term
Monic – A polynomial is monic of the leading coefficient is equal to 1
Dividing and multiplying polynomials
To divide polynomials we use the classic long division method.
To multiply polynomials it is exactly the same as expanding the brackets except there may be many more terms.
Demonstration of division and multiplication of polynomials
Factor Theorem
A factor of a polynomial is a smaller polynomial that can be divided into the original polynomials and leave a 0 remainder. If a is a root of the polynomial then (x-a) is a factor of the polynomial. Similarly if P(a) = 0 then when (x-a) is a factor.
Remainder Theorem
If a polynomial is divided by (x-a) then the remainder is P(a)
For those who are unfamiliar with the concepts of functions
Basically if P(x) = x^3 + x^2
P(1) = 1^3 + 1^2
P(-2) = (-2)^3 + (-2)^2
Sketching polynomials
A polynomial can be easily sketched if it is in factored form. Such as (x-1)(x-2) or (x+3)(x-1) (x-4). Basically to sketch we first determine all the roots. If it is a positive polynomials then start from the top right quadrant, if it is negative then start from the bottom right quadrant. Draw a curved line through all the roots and that will basically be the graph.
Repeated roots
If (x-1) is a factor of a polynomial and it is also a factor of the resulting polynomial, then (x-1) is a double root. A even root touches the x axis in the shape similar to a parabola. A odd repeated root cuts the x axis in a shape similar to a y=x^3 (cubic graph).
Finding Factors
To find factors of a polynomial it is basically a process of trial and error. All the roots of a polynomial are factors of the constant term. This is very useful information as it simplifies the trial and error process.
Anyways this should be everything that is covered in Year 10 Polynomials.
Bye
