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Solving Polynomial Equations

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A series of free, online Intermediate Algebra Lessons or Algebra II lessons.
Videos, worksheets, and activities to help Algebra students.

In this lesson, we will learn

  • how to use synthetic division to factor polynomials
  • how to solve by factoring
  • how to use the rational roots theorem
  • how to use synthetic division to solve a polynomial equation

Using Synthetic Division to Factor

One way that we can factor a polynomial is to use synthetic division. Synthetic division factoring can be an effective way to factor complicated polynomials with whole number coefficients. When factoring using synthetic division, we determine one of the roots and use synthetic division to determine the remaining coefficient. If the remaining coefficient is equal to zero, then the expression is divisible by that factor.
Polynomial Factors Found Using Synthetic Division
We are given one factor of a polynomial function and we are asked to find the other factors - which we do using synthetic division. We also find zeros and sketch the graph

Solving by Factoring

The first step is to bring all the terms to one side and set the equation equal to zero. Next, using the method of solving by factoring, take out the common terms and use one of the methods of factoring to simplify the expression. Once in this multiplication form, note that if two terms multiplied equal zero, one of the terms must be equal to zero. Given the rational roots theorem, these are the solutions to the equation.
This video explains how to solve quadratic equations by factoring.

Solving Quadratic Equations by Factoring
This video solves two quadratic equations by factoring.

Rational Roots Theorem

The rational roots theorem states that all potential roots are in the positive or negative form of the last coefficient factors divided by the first coefficient factors. With a large polynomial, solving by factoring is more difficult, and so finding the rational roots will give some potential zeros to start with. With these rational roots, the solutions after factoring complicated expressions is narrowed down to a select few answers.
How to use the rational roots theorem?

Using Synthetic Division to Solve a Polynomial Equation

To solve a polynomial equation using synthetic division, we first use the rational roots theorem to determine the potential zeros for factoring. After factoring, we can solve synthetic division polynomials by setting each of our factors equal to the other side of the equation and solving.
How to use synthetic division to solve a polynomial equation?

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