Polynomials

numberandalgebra path adv ext

Polynomials (Path) | NSW curriculum Website

• MA5-POL-P-01 defines, operates with and graphs polynomials and applies the factor and remainder theorems to solve problems

📖 Prior Knowledge

Content	Prior knowledge	Used for
Algebraic Techniques C	- expanding complex expressions - factorising quadratics	- multiplying polynomials - factorising polynomials
Non-Linear Relationships C	- sketching parabolas in factored form	- sketching polynomials in factored form
Functions and other Graphs	- function notation - transformations	- working with polynomials - transformations of polynomials

Define and operate with polynomials.pdf

• Recognise a polynomial expression $a_n{x}^n+a_{n-1}{x}^{n-1}+\dots +a_2{x}^2+a_{1}x+a_0$ where $n$=0, 1, 2… and $a_0,a_1,a_2\dots a_n$ are real numbers
• Describe polynomials using terms such as degree, leading term, coefficient and leading coefficient, constant term, monic and non-monic
• Define a monic polynomial as having a leading coefficient of one
• Apply the notation $P(x)$ for polynomials and $P(c)$ to indicate the value of $P(x)$ for $x=c$
• Add, subtract and multiply polynomials

Divide polynomials.pdf

• Identify the dividend, divisor, quotient and remainder in numerical division
• Divide a polynomial by a linear polynomial to find the quotient and remainder
• Express a polynomial in the form $P\left(x\right)=D(x)Q\left(x\right)+R(x)$, where $P(x)$ is the divisor, $Q(x)$ is the quotient and $R(x)$ is the remainder

Apply the factor and remainder theorems to solve problems.pdf

• Verify the remainder theorem and use it to find factors of polynomials and solve related problems
• Develop and apply the factor theorem to factorise particular polynomials completely and solve related problems
• Apply the factor theorem and division to find the zeroes of a polynomial $P(x)$ and solve $P(x)=0$ (degree ≤4)
• State the maximum number of zeroes a polynomial of degree $n$ can have

Graph polynomials.pdf

• Graph polynomials in factored form
• Graph quadratic, cubic and quartic polynomials by factorising and finding the zeroes
• Relate the term zeroes to polynomial functions and roots to polynomial equations
• Use graphing applications to determine the effect of single, double and triple roots of a polynomial equation $P(x)=0$ on the shape of the graph for $y=P(x)$
• Graph polynomials using the sign of the leading term and the multiplicity of roots for the equation $P(x)=0$
• Use graphing applications to compare the graphs of $y=-P(x)$, $y=P(-x)$, $y=P(x)+c$, and $y=kP(x)$ to the graph of $y=P(x)$