2.4 Polynomials

General form:

ax + by + c = 0

Standard form:

y = ax + b

From the standard form, we have:

Slope, or Gradient: a=∆y∆xa = \frac {∆y}{∆x}a=∆x∆y​, the rate of change in yyy relative to xxx.

a>0a > 0a>0: ascending rightwards

a<0a < 0a<0: descending rightwards

a=0a = 0a=0: horizontal line

yyy intercept: bbb

If l1=ax+bl_1=ax + bl1​=ax+b andl2=a′x+b′ l_2 = a'x+b'l2​=a′x+b′, then:

a=a′:l1//l2a = a': l_1 // l_2a=a′:l1​//l2​

aa′=−1:l1⊥l2aa' = -1: l_1 ⊥ l_2aa′=−1:l1​⊥l2​

Quickest way to draw this function:

Plot xxx and yyy intercepts.

Connect the two intercepts.

Find intersection by solving the simultaneous equation. If

l_1=ax + b

and

l_2 = a'x+b'

, solve:

The formula to find a linear function given a point (

x_0, y_0

) and slope

a

is:

y -y_0 = a(x-x_0)

Figure 2.4.1 Graphical representation of a linear equation

Quadratics

General form:

y =ax^2+bx+c

Vertex form:

y =a(x-b)^2+c

Intercept form:

y =a(x-b)(x-c)

From the standard form, we have:

Axis of symmetry: x=−b2ax = -\frac b{2a}x=−2ab​

Vertex: (−b2a,f(−b2a)-\frac b{2a}, f(-\frac b{2a})−2ab​,f(−2ab​))

yyy-intercept: ccc

For the sign of aaa,

a>0a > 0a>0 then concave up

a<0a < 0a<0 then concave down

Vertex marks the minimum/maximum of the graph depending on the concavity.

Quickest way to draw this function:

Convert any other form to the vertex form: y=a(x−b)2+cy =a(x-b)^2+cy=a(x−b)2+c

Plot the vertex (b,cb, cb,c)

Find the yyy-intercept.

Connect the vertex and yyy-intercept with consideration of concavity.

To solve for the quadratic equation:

Use (a+b)2=a2+2ab+b2(a+b)^2=a^2+2ab+b^2(a+b)2=a2+2ab+b2 to complete the square.

Factorize where possible.

Apply the quadratic formula.

Discriminant

∆ = b^2-4ac

has three cases:

∆>0∆ > 0∆>0: two real conjugate roots

∆=0∆ = 0∆=0: repeated roots

∆<0∆ < 0∆<0: two imaginary conjugate roots

Figure 2.4.2 Quadratic function with two distinct real roots

Polynomials of degree

n

(HL)

General form:

P(x)=a_nx^n+a_{n-1}x^{n-1}+ ... + a_0

Degree: the highest power of the polynomial

We have several theorems for general polynomials:

Theorem	Features
Euclidean Algorithm	We have $f$ and $Q$ such that $P(x)=f(x)Q(x)+R(x)$ . Notice that $deg(R)=0$ or $deg(R)<deg(Q)$
Remainder theorem	If we divide $P(x)$ by $(x-c)$ , then $P(c)$ is the remainder.
Factor theorem	If $(x-c)$ divides $P(x)$ , then $P(c)=0$
Fundamental theorem of Algebra	Over the field ℂ, the polynomial $P(x)$ of degree $n$ has $n$ roots in ℂ. The complex roots appear as conjugates, i.e. $a ± bi$ . It follows that odd degree polynomial $P(x)$ has at least one real root.
Vieta’s theorem	Let the $n$ roots of $P(x)$ be $\alpha_1 ... \alpha_n$ . Then, we have: 1. Sum of all roots: $\sum^n_{i=0}\alpha_i=-\frac{\alpha_{n-1}}{\alpha_n}$ 2. Product of all roots: $\Pi^n_{i=0}\alpha_i=\frac{\alpha_0}{\alpha_n}$

Modulus functions

|x|

denotes the absolute value of

x

, the distance from 0 in the number line.

Some important properties are listed below:

∣x∣=x2|x|=\sqrt{x^2}∣x∣=x2​

∣x∣2=x2|x|^2=x^2∣x∣2=x2

∣xy∣=∣x∣∣y∣|xy|=|x||y|∣xy∣=∣x∣∣y∣

∣x∣=∣a∣|x|=|a|∣x∣=∣a∣ or ∣x∣=a⇒x=±a|x|=a \Rightarrow x=±a∣x∣=a⇒x=±a

∣x+y∣≤∣x∣+∣y∣|x + y| ≤ |x| + |y| ∣x+y∣≤∣x∣+∣y∣ *Triangle inequality

These are different forms of modulus function:

Features

Function

y=|f(x)|

Reflect the region

f(x)<0

by the

x

axis.

y=f(|x|)

Reflect the region

x \geq 0

by the

y

axis.

|y|=f(x)

Reflect the region

y\geq0

by the

x

axis.

|y|=f(|x|)

Draw the graph in the 1st quadrant, and reflect on all remaining quadrants.