#Functions #Year11 #Ext1 >[!info]- [Polynomials | NSW Curriculum Website](https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-extension-1-11-12-2024/content/n11/fa858a6f68) >- ME1-11-02 applies the remainder and factor theorem and sums and products of zeroes to solve problems involving polynomials ## 📖 Prior Knowledge | Content | Prior knowledge | Used for | | ------------------------------------------ | ------------------------------ | -------------------------- | | [[Working with Functions]] | - function notation | - working with polynomials | | [[Stage 5 Paths/Polynomials\|Polynomials]] | - factor and remainder theorem | - *repeated content* | ## Language and graphs of polynomials - Define a polynomial function $P(x)$ of degree $n$, where $n$ is a non-negative integer, to be a function that can be expressed in the form $P(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+a_2x^2+a_1x+a_0$, for real $a_n,\ldots,a_0$ and $a_n \neq 0$ - Define the leading term of $P(x)$ to be the term of highest degree and define the leading coefficient of $P(x)$ to be the coefficient of the leading term - Define the constant term of $P(x)$ to be $a_0$ - Define a polynomial to be monic if its leading coefficient is 1 - Define the zero polynomial to be $P(x)=0$ to be the polynomial with all the coefficients equal to zero and recognise that the zero polynomial has no leading term, no leading coefficient, no degree and constant term 0 - Determine the degree of $P(x)+Q(x)$ when two non-zero polynomials $P(x)$ and $Q(x)$, of degrees $n$ and $m$ respectively, are added - Explain how the leading coefficient and the degree determine whether $y \rightarrow \infty$ or $y \rightarrow -\infty$ as $x \rightarrow \infty$ and as $x \rightarrow -\infty$ - Define the zeroes of $P(x)$ to be the numbers $\alpha$ such that $P(\alpha)=0$, and define the roots of the polynomial equation $P(x)=0$ to be its solutions, and recognise that every real number is a zero of the zero polynomial - Define $\alpha$ as a repeated zero or multiple zero of a non-zero polynomial $P(x)=(x-\alpha)Q(x)$ when $(x-\alpha)$ is a factor of $Q(x)$, and $\alpha$ as a single zero of $P(x)$ when $(x-\alpha)$ is not a factor of $Q(x)$, for $Q(x) \neq 0$ - Define $\alpha$ as a zero of $P(x)$ of multiplicity $m$ if $P(x)=(x-\alpha)^mQ(x)$, where $m$ is a positive integer and $Q(\alpha) \neq 0$ - State the multiplicity of each root of a polynomial equation given in factored form - Find the zeroes of a polynomial that is expressed as a product of linear factors, determine their multiplicity and graph the polynomial ## Remainder and factor theorems - Examine the process of division of polynomials by comparing with the process of division with remainders for whole numbers, and use the terms dividend, divisor, quotient and remainder - Express in the form $P(x)=A(x)Q(x)+R(x)$ the result of dividing $P(x)$ by a divisor $A(x)$, that is not the zero polynomial, with quotient $Q(x)$ and remainder $R(x)$, and explain why either $R(x)=0$ or $\deg{R(x)}<\deg{A(x)}$ - Express the result of the division also in the form $\frac{P(x)}{A(x)}=Q(x)+\frac{R(x)}{A(x)}$ - Explain why division by $x-\alpha$ yields $P(x)=(x-\alpha)Q(x)+r$, where $r$ is a constant, and why $x-\alpha$ is a factor if and only if $r=0$ - Prove and apply the remainder theorem for polynomials: when $P(x)$ is divided by $x-\alpha$, the remainder is $P(\alpha)$, and solve related polynomial problems - Prove and apply the factor theorem for polynomials: $P(\alpha)=0$ if and only if $x-\alpha$ is a factor of $P(x)$, and solve related polynomial problems - Use the factor theorem to find all factors of $P(x)$ of the form $x-\alpha$, where $\alpha$ is an integer, and use division to find the remaining factor of the polynomial ## Sums and products of zeroes of polynomials - Prove that if a quadratic $P(x)=ax^2+bx+c$ has zeroes $\alpha$ and $\beta$ then $\alpha+\beta=-\frac{b}{a}$, the sum of the zeroes, and $\alpha\beta=\frac{c}{a}$, the product of the zeroes - Prove that if a cubic $P(x)=ax^3+bx^2+cx+d$ has three zeroes $\alpha$, $\beta$, $\gamma$, then $\alpha+\beta+\gamma=-\frac{b}{a}$, $\alpha\beta+\beta\gamma+\gamma\alpha=\frac{c}{a}$ and $\alpha\beta\gamma=-\frac{d}{a}$ - Prove that if a quartic $P(x)=ax^4+bx^3+cx^2+dx+e$ has four zeroes $\alpha$, $\beta$, $\gamma$, $\delta$ then $\alpha+\beta+\gamma+\delta=-\frac{b}{a}$, $\alpha\beta+\alpha\gamma+\alpha\delta+\beta\gamma+\beta\delta+\gamma\alpha=\frac{c}{a}$, $\alpha\beta\gamma+\beta\gamma\delta+\gamma\delta\alpha+\delta\alpha\beta=-\frac{d}{a}$ and $\alpha\beta\gamma\delta=\frac{e}{a}$ - Use the formulas for the sums and products of zeroes to solve problems involving zeroes and coefficients of quadratic, cubic and quartic polynomials