Pre-Calculus

Notes from my pre-calculus course in NCHU

What is the degree of this polynomial?
3x + 2

1, linear

What is the degree of this polynomial?
3x² + 2

2, quadratic

What is the degree of this polynomial?
3x³ + 2

3, cubic

What is the degree of this polynomial?
3x⁴ + 2

4, quartic

Here is a polynomial:
9 + 4x + 3x²
Name the different parts of it

9 = constant
4 = coefficient
3 = leading coefficient
² = exponent

Write the following polynomial in standard form:
2x + 18 - 9x² - 78x⁴

- 78x⁴ - 9x² + 2x + 18

Write the Distributive Law

a(b+c) = ab + ac

Differentiate distributing and factoring

Distributing breaks open the parentheses.
Factoring creates the parentheses.

What is a rational expression?

A fraction where both the numerator and denominator are polynomials.

Can x = 1?

6
x-1

No, because division by 0 is not possible.

Can m = 3 / Can m = -2 ?

m⁴ + 18m + 2
m² - m - 6

No, because division by 0 is not possible.

What is the difference of squares formula for factoring?

a² - b² = (a - b)(a + b)

Note: not applicable to a² + b².

What is this?

General formula for dividing rational expressions

What are these?

General formulae for adding/subtracting rational expressions.

What needs to happen for adding/subtracting fractions?

There needs to be a common denominator

What formula can you use to check if an expression can be factored further?

Use the discriminant: b² - 4ac
If positive, it can be factored into brackets.
If = 0, it is a perfect square trinomial = can be factored.
If negative, it cannot be factored.

Division of rational expressions

What is the term given to a polynomial that cannot be factored?

Prime polynomial

What is the zero-factor property?

If ab = 0 then either a = 0 or b = 0

Formula to find a slope

m = y2 - y1 over x2 - x1

Slope-intercept form

y = mx + b (b = y-intercept)

Point-slope form

y - y1 = m(x - x1)

Standard form for a line

ax + by = c

Slope and equation of a horizontal line

m = 0

example equation: y = 18

Slope and equation of a vertical line

m = undefined

example equation: x = 7

How to find the perpendicular slope

Flip the fraction and change the sign.

Eg. if the slope is 2, the slope of the perpendicular line is -1/2