Polynomial Functions [Video Tutorial]

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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