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
Worksheet
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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