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Polynomial Functions

This tutorial covers material encountered in chapter 4 of the VCE Mathematical Methods Textbook, namely:

  • The turning point form and axis of symmetry of a quadratic polynomial
  • The quadratic formula and the discriminant (extremely useful!)
  • Dividing polynomials via long division and equating coefficients
  • The Remainder and Factor theorem
  • Study of polynomials in general

Q1 – Quadratic Polynomials and Graphs

Q2 – Cubic Polynomials and Graphs

Q3 – Polynomials in Turning Point Form

Q4 – Dividing Polynomial (the Remainder Theorem)

Q5 – Polynomial Long Division, Quotient and Remainder

Q6 – Polynomials and Solutions

Q7 – Find the Rule of a Polynomial Function


Q1. Sketch the graph of each of the following polynomials, clearly indicating the axis intercepts and the coordinates of the vertex:

(a) h(x)=2(x-4)^2+1

(b) j(x)=x^2-3x+6

(c) b(x)=x^2-1

(d) c(x)=3x^2+9x-7

Q2. Sketch the graph of each of the following cubic polynomials, clearly indicating the axis intercept(s) and the coordinates of the zero gradient:

(a) f(x)=2(x-1)^3-16.

(b) h(x)=3(x-4)^3+7

(c) g(x)=-2(x+2)^3-9

Q3. Express each of these polynomials in turning point form:

(a) x^2+2x

(b) x^2-6x+8

(c) 2x^2+4x-9

(d) -x^2+3x-4

Q4. Without dividing, find the remainder when the first polynomial is divided by the second (Use the Remainder theorem!)

(a) x^3+2x^2+5x+1,\,x+1

(b) x^3-3x^2-x+6,\,x-2

(c) -2x^3+x^2+5x,\,3x+2

Q5. What is the quotient and remainder when x^4+3x^3+2x^2+x+1 is divided by x^2-2x+2 ?

Q6. For what values of l\in \R does p(x)=2x^2-2lx+l+5 have no real solutions?

Q7. The function f:\R\to\R,\,f(x) is a polynomial function of degree 4. Part of the graph of f is shown below, with its x intercepts labelled. If f(1) = 10 find the rule of f.

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