11–12Mathematics Extension 1 11–12 Syllabus

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Year 11

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Polynomials

MAO-WM-01
develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly
ME1-11-02
applies the remainder and factor theorem and sums and products of zeroes to solve problems involving polynomials

Polynomials

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 $a_{n} \neq 0$
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 $y \to \infty$ or $y \to - \infty$ as $x \to \infty$ and as $x \to - \infty$
Define the zeroes of $P (x)$ to be the numbers $α$ such that $P (α) = 0$ , 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 $α$ as a repeated zero or multiple zero of a non-zero polynomial $P (x) = (x - α) Q (x)$ when $(x - α)$ is a factor of $Q (x)$ , and $α$ as a single zero of $P (x)$ when $(x - α)$ is not a factor of $Q (x)$ , for $Q (x) \neq 0$
Define $α$ as a zero of $P (x)$ of multiplicity $m$ if $P (x) = {(x - α)}^{m} Q (x)$ , where $m$ is a positive integer and $Q (α) \neq 0$
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 $x - α$ yields $P (x) = (x - α) Q (x) + r$ , where $r$ is a constant, and why $x - α$ is a factor if and only if $r = 0$
Prove and apply the remainder theorem for polynomials: when $P (x)$ is divided by $x - α$ , the remainder is $P (α)$ , and solve related polynomial problems
Prove and apply the factor theorem for polynomials: $P (α) = 0$ if and only if $x - α$ 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 $x - α$ , where $α$ 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 $α$ and $β$ then $α + β = - \frac{b}{a}$ , the sum of the zeroes, and $αβ = \frac{c}{a}$ , the product of the zeroes
Prove that if a cubic $P (x) = a x^{3} + b x^{2} + cx + d$ has three zeroes $α$ , $β$ , $γ$ , then $α + β + γ = - \frac{b}{a}$ , $αβ + βγ + γα = \frac{c}{a}$ and $αβγ = - \frac{d}{a}$
Prove that if a quartic $P (x) = a x^{4} + b x^{3} + c x^{2} + dx + e$ has four zeroes $α$ , $β$ , $γ$ , $δ$ then $α + β + γ + δ = - \frac{b}{a}$ , $αβ + αγ + αδ + βγ + βδ + γα = \frac{c}{a}$ , $αβγ + βγδ + γδα + δαβ = - \frac{d}{a}$ and $αβγδ = \frac{e}{a}$
Use the formulas for the sums and products of zeroes to solve problems involving zeroes and coefficients of quadratic, cubic and quartic polynomials

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Year 12

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11–12Mathematics Extension 1 11–12 Syllabus

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