Free Algebra
Miscellaneous Equations
Operations with Fractions
Undefined Rational Expressions
Writing Equations for Lines Using Sequences
Intersections of Lines and Conics
Graphing Linear Equations
Solving Equations with Log Terms and Other Terms
Quadratic Expresions - Complete Squares
Adding and Subtracting Fractions with Like Denominators
Multiplying a Fraction by a Whole Number
Solving Equations with Log Terms and Other Terms
Solving Quadratic Equations by Factoring
Locating the Solutions of the Quadratic Equation
Properties of Exponents
Solving Equations with Log Terms on Each Side
Graphs of Trigonometric Functions
Estimating Products and Quotients of Mixed Numbers
The circle
Adding Polynomials
Adding Fractions with Unlike Denominators
Factoring Polynomials
Linear Equations
Powers of Ten
Straight Lines
Dividing With Fractions
Multiplication Property of Equality
Rationalizing Denominators
Multiplying And Dividing Fractions
Distance Between Points on a Number Line
Solving Proportions Using Cross Multiplication
Using the Quadratic Formula
Scientific Notation
Imaginary Numbers
Values of Symbols for Which Fractions are Undefined
Graphing Equations in Three Variables
Writing Fractions as Decimals
Solving an Equation with Two Radical Terms
Solving Linear Systems of Equations by Elimination
Factoring Trinomials
Positive Rational Exponents
Adding and Subtracting Fractions
Negative Integer Exponents
Rise and Run
Multiplying Square Roots
Multiplying Polynomials
Solving Systems of Linear Inequalities
Multiplication Property of Radicals
A Quadratic within a Quadratic
Graphing a Linear Equation
Calculations with Hundreds and Thousands
Multiplication Property of Square and Cube  Roots
Solving Equations with One Log Term
The Cartesian Coordinate Plane
Equivalent Fractions
Adding and Subtracting Square Roots
Solving Systems of Equations
Exponent Laws
Solving Quadratic Equations
Factoring Trinomials
Solving a System of Three Linear Equations by Elimination
Factoring Expressions
Adding and Subtracting Fractions
The parabola
Computations with Scientific Notation
Quadratic Equations
Finding the Greatest Common Factor
Introduction to Fractions
Simplifying Radical Expressions Containing One Term
Polynomial Equations
Graphing and Intercepts
The Number Line
Adding and Subtracting Rational Expressions with Different Denominators
Scientific Notation vs Standard Notation
Factoring by Grouping
Extraneous Roots
Variables and Expressions
Linera Equations
Integers and Substitutions
Squares and Square Roots
Adding and Subtracting Rational Expressions with Different Denominators
Solving Linear Inequalities
Expansion of a Product of Binomials
Powers and Exponents
Finding The Greatest Common Factor
Quadratic Functions
The Intercepts of a Parabola
Solving Equations Containing Rational Expressions
Subtracting Polynomials
Solving Equations
Adding Fractions with Unlike Denominators
Solving Systems of Equations by Substitution
Solving Equations
Product and Quotient of Functions
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Polynomial Equations

Solving a Quadratic Inequality

Solve the quadratic inequality:  12· x2 - 22 · x 4.


The graphs of the quadratic (  y =12 · x2 - 22· x) and the horizontal line ( y = 4) are shown together in the graph below. As we are looking for the x-values that make the height of the quadratic less than the height of the horizontal line, our answer will consist of all xvalues that lie between the two intersection points (including the intersection points themselves as the “” sign is used, rather than the “<” sign.) 

To locate the x-coordinates of the points where the quadratic and the horizontal line meet, we must manipulate the given inequality to make the right hand side of the inequality equal to zero. To do this we can subtract 4 from both sides. 

12 · x2 - 22 · x - 4 0. (Subtract 4 from both sides)

To find the x-coordinates of the intersection points, you need to solve the quadratic equation: 

12 · x2 - 22 · x - 4 = 0.

Solving this quadratic equation with the quadratic formula gives:

So, the solution to the quadratic inequality consists of the intervals of x-values: 

Polynomial Equations

A polynomial equation is an equation of the form:

y = c0 + c1·x + c2·x2 + ... + cn·xn

where the powers of x must be positive integers. The letters c0, c1, … , cn represent numbers.

The largest power of x in the polynomial is called the degree of the polynomial.


Polynomial Functions that You Already Know About

Each of the different types of polynomial functions has a distinctively shaped graph. Some of the polynomials (and shapes) that you are already familiar with include:

Degree 0 polynomials (i.e. constant functions) have graphs that are horizontal lines.

Degree 1 polynomials (i.e. linear functions) have graphs that are straight lines.

Degree 2 polynomials (i.e. quadratic functions) have graphs that are parabolas.

Degree 3 polynomials (called cubics) have graphs that look like John Travolta from the cover of the Saturday Night Fever movie soundtrack (see Figure 1).

Figure 1: The graph of a polynomial function of degree 3.


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