Thinking Question

Why do we spend so much time studying quadratic functions, but relatively little time on cubic functions, quartic functions, quintic functions, and so on?

A syntactic property of equation solving

Pay attention to how many times the variable appears in an equation.

Principle 1: If a variable appears only once in an equation, then we can solve for x relatively easily (by applying various inverse operations to both sides).

Principle 2: To solve an equation, we must transform it so that the variable appears only once.

Example

For each equation, state how many times the variable appears and then solve the equation. Pay special attention to any step where the number of times the variable appears is reduced.

  1. 2 x+7=3
  2. 7 x-2 x+11=-20
  3. e^{3 x}+5=7
  4. \sqrt{\frac{2 x-5}{3}}=4
  5. (x+3)(x-2)=0
  6. (x+3)(x+5)=24
  7. (x+4)^{2}=25

What kinds of mathematical operations or steps actually reduce the number of times a variable appears in an equation? Make a list.

Why quadratics?

[T]he study of quadratic equations is only a small part of the study of quadratic functions; namely, the former is about how to locate the zeros of quadratic functions. Moreover, TSM makes this topic more difficult than it needs to be, partly by presenting the technique of completing the square as a rote skill for one purpose only: getting the quadratic formula. Consequently, the quadratic formula ends up also being a rote skill and, likewise, the formula for the vertex of the graph of a quadratic function. In Section 2.1, however, we show that completing the square is the major idea that 

(a) leads to the proof of the quadratic formula and the formula of the vertex of the graph (see page 75),

(b) proves that the graph of f(x) = ax2+bx+c is congruent to the graph of fa(x) = ax2, 

(c) exhibits the commonality between the study of linear and quadratic functions, namely, the fact that both revolve around the shape of the graphs of the representative functions ax and ax2 (see page 73)

Wu, A&G, Preface pxiii

After such a detailed study of linear and quadratic functions, one would expect another long chapter on cubic functions (polynomial functions of degree 3) and yet another on quartic functions (polynomial functions of degree 4), etc. The fact that this does not happen is a consequence of the fact that the theory loses its simplicity when the degree of the polynomial exceeds 2. Recall that the study of quadratic functions was greatly facilitated by the method of completing the square. There is no tool of comparable power and simplicity for polynomial functions of degree exceeding 2. Analogs of the quadratic formula for quadratic equations continue to hold for cubic and quartic equations,  but they are unwieldy and therefore not particularly useful. It is a famous theorem of Abel and Galois that for equations of degrees exceeding 4, no analog of the quadratic formula exists. Consequently, we know far less about arbitrary polynomial functions. All we can do is to give a general discussion of the most basic properties of polynomial functions and rational functions.

Wu, A&G, p121

Completing the square

If we give ourselves the power to add a constant to our expression, then we can always turn the sum of an x^2 term and an x term, ax^2+bx, into a perfect square.

Demonstration

Multiplication as area.

  1. 3\cdot5  (rectangle)
  2. a\cdot b (rectangle)
  3. x^2 (square)
  4. (x+5)^2 (square)
  5. (x+k)^2 (square – equivalent to sum of 4 rectangles)

Rewrite each expression using only a single instance of the variable x:

  1. x^2+6x
  2. x^2-12x
  3. x^2+5x
  4. x^2+8x+11
  5. x^2-20x+16
  6. 3x^2+12x-21
  7. 7x^2-5x+12

Polynomials and roots

What are the polynomials? They are a particular collection of functions.

The idea of polynomials

The constant functions and the function f(x)=x are polynomials. If you combine two polynomials using the operations addition, subtraction, and multiplication, then the result is a polynomial.

The definition of polynomials

Definition: a polynomial function is any function of the form f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0} for fixed numbers a_{n}, \ldots, a_{0} \left(a_{n} \neq 0\right) and n is called its degree.

Theorem. If a polynomial f(x) has a root r, then f(x) can be written in the form f(x)=(x-r)\cdot q(x), where q(x) has degree one less than the degree of f(x).

What does this theorem allow us to say about the roots of a polynomial f(x) of degree 4? Discuss

Theorem. If a polynomial f(x) has odd degree, then it has at least one root.

Think about:

  • Can all polynomials be factored (into linear factors)?
  • Does every polynomial have a root?
  • Is it possible to factor a polynomial that has no root? (ex: x^4+3x^2+2)