4.2 Quadratic and cubic polynomials

A polynomial of degree $n$

has $n$ roots, which may be real or complex. Real roots may be distinct or repeated. Complex roots always come in conjugate pairs, $a \pm bi$, where a is called the real part, $\pm b$ is called the imaginary part, and $i = \sqrt{-1}$ is the imaginary unit.

Quadratic polynomials¶

Quadratic polynomials can have three types of roots:

• Two real roots

• One (repeated) root

• Two complex roots

Graphically, two real roots occur as two intersections of a parabola with the y-axis. For example, the roots of $x^2 - 1 = 0$ occur at $x = \pm 1$:

One repeated root occurs as a single point of a parabola (its minimum or maximum) touching the y-axis. For example, the root of $(x-1)^2 = 0$ occurs at $x = 1$:

Complex roots do not appear as intersections on a graph. For example, the roots of $x^2 + 1 = 0$ are $x = \pm i$:

There are several strategies that can be used to find roots of quadratic polynomials:

• Factoring

  $$x^2 + 8 x - 9 = (x+9)(x-1) = 0$$

  (2)

  This equation will be true if either factor is zero, so the roots are $x = -9$ or $x = 1$.

• Completing the square

  $$\begin{align} x^2 + 8x &= 9 \\ x^2 + 8x + 16 &= 9 + 16 \\ (x+4)^2 &= 25 \\ x + 4 &= \pm \sqrt{25} x = -4 \pm 5 \end{align}$$

  (3)

  This gives the same roots!

• Quadratic formula

  For the polynomial $x^2 + 8x - 9 = 0$,

  $$\begin{align} x &= \frac{-8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot -9}}{2 \cdot 1} \\ &= \frac{-8 \pm \sqrt{100}}{2} \\ &= -4 \pm 5 \end{align}$$

  (5)

Example: Box optimization¶

An open-top box will be made from an 8.5" x 11" piece of paper by cutting out a square from each corner and folding the flaps. What size square should be cut to make the biggest box?

The volume of the box that will be obtained by cutting a square of edge length x is:

The volume will be at an extremum with respect to x if $V'(x) = 0$:

The roots of this quadratic polynomial are:

Only the first root, 1.59", is physical because there is not enough material to remove 4.91" from two corners!

Example: Complex roots¶

Find the roots of

I will use the technique of completing the square:

Cubic (and higher) polynomials¶

General formulas for roots of cubic and quartic polynomials are known but complicated. No such formula exists for higher-order polynomials. In these cases, numerical methods are needed!

Skill builder problems¶

Solve all roots to 3 significant figures.