Polynomials

functions year11 ext1

Polynomials | NSW Curriculum Website

• ME1-11-02 applies the remainder and factor theorem and sums and products of zeroes to solve problems involving polynomials

📖 Prior Knowledge

Content	Prerequisite relationships
Algebraic Techniques	- Substitute values into algebraic expressions → Use polynomial notation P(x)
Indices	- Describe numbers in index form → Define and describe polynomials
Algebraic Techniques A	- Expand binomial products → Add, subtract and multiply polynomials
Algebraic Techniques C	- Factorise using special products and grouping → Apply the factor theorem
Equations C	- Solve quadratic equations by factorisation → Sum and product of roots of a quadratic
Non-Linear Relationships C	- Graph and compare power curves → Graph polynomials from factored form
Polynomials	- Define and describe polynomials (revisited content) - Use polynomial notation P(x) (revisited content) - Add, subtract and multiply polynomials (revisited content) - Divide a polynomial by a linear polynomial (revisited content) - Apply the remainder theorem (revisited content) - Apply the factor theorem (revisited content) - Graph polynomials from factored form (revisited content) - Graph polynomials using roots and leading term (revisited content) - Remainder on division by a quadratic divisor (revisited content) - Find the equation of a polynomial from its graph (revisited content)
Functions and other Graphs	- Use function notation f(x) → Use polynomial notation P(x)

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 $P(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\dots +a_2{x}^2+a_{1}x+a_0$, for real $a_n,\dots ,a_0$ and $a_n\ne 0$
• 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 $y\to \infty$ or $y\to -\infty$ as $x\to \infty$ and as $x\to -\infty$
• 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 $Q(x)\ne 0$
• Define $\alpha$ as a zero of $P(x)$ of multiplicity $m$ if $P(x)=(x-\alpha {)}^{m}Q(x)$, where $m$ is a positive integer and $Q(\alpha )\ne 0$
• 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 $\mathrm{deg}R(x)<\mathrm{deg}A(x)$
• 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 $P(x)=a{x}^2+bx+c$ 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 $P(x)=a{x}^3+b{x}^2+cx+d$ 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 $P(x)=a{x}^4+b{x}^3+c{x}^2+dx+e$ 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