NSW Y11 Maths - Extension 1 Polynomials Remainder Theorem

Resources for Remainder Theorem

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Remainder Theorem Theory

NSW Syllabus Reference

NSW Syllabus Reference: ME-F2.1: Remainder and factor theorems. This will require student to 

  • define a general polynomial in one variable, \(x\), of degree \(n\) with real coefficients to be the expression: \(a_{n}x^n+a_{n-1}x^{n-1}+\cdots +a_{2}x^{2}+a_{1}x+a_{0}\), where \(a_{n}\neq 0\)
  • use division of polynomials to express \(P(x)\) in the form \(P(x)=A(x).Q(x)+R(x)\) where \(\deg R(x)<\deg A(x)\) and \(A(x) \) is a linear or quadratic divisor, \(Q(x)\) the quotient and \(R(x)\) the remainder
  • prove and apply the factor theorem and the remainder theorem for polynomials and hence solve simple polynomial equations (ACMSM089, ACMSM091)

Ref: https://educationstandards.nsw.edu.au/

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Videos

Videos relating to Remainder Theorem.

  • Remainder Theorem - Video - Remainder Theorem

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  • Remainder Theorem - Video - Remainder Theorem 2

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  • Remainder Theorem - Video - Remainder Theorem and Synthetic Division of Polynomials

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Theory

If a polynomial \(P(x)\) is divided by \((x - a)\) until the remainder \(R\) does not contain \(x\), then \(R = P(a).\)

For example: \(P(x) = (x - 2)Q(x) + R,\,\,P(2) = (2 - 2)Q(x) + R \to R = P(2)\)

Note that \(P(x) = (x + 2)Q(x) + R,\,\,P( - 2) = ( - 2 + 2)Q(x) + R \to R = P( - 2)\)

Syllabus Reference

NSW Syllabus Reference: ME-F1.1: Graphical relationships. This will require student to 

  • examine the relationship between the graph of \(y=f(x)\) and the graph of \(y=\dfrac{1}{f(x)}\) and hence sketch the graphs (ACMSM099)
  • examine the relationship between the graph of \(y=f(x)\) and the graphs of \(y^2=f(x)\) and \(y=\sqrt{f(x)}\) and hence sketch the graphs
  • examine the relationship between the graph of \(y=f(x)\) and the graphs of \(y=|f(x)|\) and \(y=f(x)+g(x)\) and hence sketch the graphs (ACMSM099)
  • examine the relationship between the graphs of \(y=f(x)\) and \(y=g(x)\) and the graphs of \(y=f(x)+g(x)\) and \(y=f(x)g(x)\) and hence sketch the graphs
  • apply knowledge of graphical relationships to solve problems in practical and abstract contexts

Ref: https://educationstandards.nsw.edu.au/