Solving Quadratic (and polynomial) Equations by Factoring

For any real number a and b, if ab=0 then either a = 0, b = 0, or both equal 0. Examples: When solving polynomial equations, factor the expression and set each factor equal to zero. Zero Product Property

In this example, the equation is already factored and is set equal to zero. To solve, apply the Zero Product Property by setting each factor equal to zero. If then and Solve the factored equations or The solutions are x = -3 and x = ½ or { x | x = -3, ½} Solving Quadratic (and polynomial) Equations Subtract 3Add 1, Divide by 2

Solve the following quadratic equation. Start by factoring the quadratic expression. Factor using difference of squares Set each factor equal to zero and solve each equation. or Solving Quadratic (and polynomial) Equations

Factor the trinomial. or Solving Quadratic (and polynomial) Equations

Solve the following quadratic equations. 1) 2) 3) 4) 5) and Linear equations have only 1 solution

Let’s look at some of the graphs…. x 2 + x -12 = 02x 2 + 8x = 0 4x 2 = 25

Solve the following Cubic Equation Factor out the GCF. Factor the trinomial. or

Check out the graph….. or

Solve the Quartic Equation Put it in standard form Factor – how low can you go? We’re not done yet!!

Set each factor equal to zero. Solve the Quartic Equation Why don’t we need to set the factor of 4 equal to zero? Now solve each factor for x.

Solving by factoring, we found the zeros of 4x 4 = 64 to be x = ±2, ±2i. When factoring we can sometimes find imaginary solutions along with the real ones. Notice – what degree was our polynomial? How many solutions did we find? Solve the Quartic Equation 4th 4!!

Roots of Equations  In the last example we found two types of roots – real and imaginary.  Real roots can be seen on the graph – the curve crosses the x -axis at those values.  Imaginary roots cannot been seen on the graph. Many functions have imaginary solutions.  For polynomial functions, the degree tells you the total number of roots – real and imaginary combined!

Roots of Equations  Thus a cubic equation should have how many roots?  A quintic equation should have how many roots?  Remember, all the roots may not be REAL, some could be imaginary. 3 5