Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Quadratic Functions
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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Solving Quadratic and Polynomial Equations

When you solve a quadratic equation when you have been given a y-value and need to find all of the corresponding x-values. For example, if you had been given the quadratic equation:

y = x2 + 8 · x +10,

and the y-value,

y = 30,

then solving the quadratic equation would mean finding all of the numerical values of x that work when you plug them into the equation:

x2 + 8 · x +10 = 30.

Note that solving this quadratic equation is the same as solving the quadratic equation:

x2 + 8 · x +10 - 30 = 30 - 30 (Subtract 30 from each side)
x2 + 8 · x - 20 = 0 (Simplify)

Solving the quadratic equation x2 + 8 · x - 20 = 0 will give exactly the same values for x that solving the original quadratic equation, x2 + 8 · x +10 = 30, will give.

The advantage of manipulating the quadratic equation to reduce one side of the equation to zero before attempting to find any values of x is that this manipulation creates a new quadratic equation that can be solved using some fairly standard techniques and formulas.

Solving a polynomial equation is exactly the same kind of process as solving a quadratic equation, except that the quadratic might be replaced by a different kind of polynomial (such as a cubic or a quartic).


The Number of Solutions of a Polynomial Equation

A quadratic is a degree 2 polynomial. This means that the highest power of x that shows up in a quadratic’s formula is x2. The maximum number of solutions that a quadratic function can possibly have is 2.

The maximum number of solutions that a polynomial equation can have is equal to the degree of the polynomial.

It is possible for a polynomial equation to have fewer solutions (or none at all). The degree of the polynomial gives you the maximum number of solutions that are theoretically possible, not the actual number of solutions that will occur.

Example: Solving a Polynomial Equation Graphically

The graph given below shows the graph of the polynomial function:

Use the graph to find all solutions of the polynomial equation:


Graphically, as the polynomial equation is equal to zero the solutions of the polynomial equation,

will be the x-coordinates of the points where the graph of the polynomial:

touches or crosses the x-axis. If you look carefully at the graph supplied above, the graph of the polynomial touches or cuts the graph at the following points:

x = -2, x = -1, x = 2.

The solutions of the polynomial equation

are x = -2, x = -1 and x = 2.


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