#NumberAndAlgebra #Path #Adv #Ext >[!info]- [Polynomials (Path) | NSW curriculum Website](https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-k-10-2022/content/stage-5/fac28d3aef) >- 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<br>- factorising quadratics | - multiplying polynomials<br>- factorising polynomials | | [[Non-Linear Relationships C]] | - sketching parabolas in factored form | - sketching polynomials in factored form | | [[Functions and other Graphs]] | - function notation<br>- transformations | - working with polynomials<br>- transformations of polynomials | ## [[Define and operate with polynomials.pdf]] - Recognise a polynomial expression $a_n\ x^n+a_{n\ -\ 1}\ x^{n\ -\ 1}+\ldots+a_2\ x^2+a_1\ x+a_0$ 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 $y=P(x)$