Why learn how to factor?
Moving Madness
The way you organize your belongings when you move can have a huge impact on the success of your move. For example, you wouldn’t want to try and move all your things using only shoeboxes. Not only would it take forever, you probably own things that are larger than a shoebox. Storing and sorting the nuts and bolts in your toolshed is very different from storing and sorting all of your possessions when you move.
In this module, we will present some factoring techniques for polynomials that will help you solve polynomial equations. Factoring is a complementary operation to the distributive property, it is a way to “unpack” the multiplication done by applying the distributive property. Reorganizing polynomials by factoring allows us to find solutions for certain types of polynomials.
You may be thinking, we already know how to solve linear equations, why don’t we just use the same techniques for solving polynomial equations? Let’s investigate the example below to see why we need different techniques for solving polynomial equations than for solving linear equations.
Solve the following second degree polynomial equation: [latex]x^2+6x-7=2x-4[/latex]
We use complementary operations to isolate the variable when we solve linear equations, so we will try to do the same with this polynomial equation.
[latex]\begin{array}{l}x^2+6x-7=2x-4\\\,\,\,\,\,\,\,\,\underline{-2x}\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{-2x}\,\,\,\,\,\,\\x^2+4x-7=-4\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{+7}\,\,\,\,\,\,\,\,\,\underline{+7}\\x^2+4x=3\end{array}[/latex]
Ok, now what? How can we isolate [latex]x[/latex] with the [latex]x^2[/latex] term in the problem? We will need different techniques than we use for solving linear equations to solve this equation. There are many methods for solving different kinds of polynomials using algebraic principles, but the truth is that most polynomials cannot be solved with the algebra we have used here.
At the end of this module, we will share some of the practical uses for solving polynomials that occur in our everyday lives. Polynomials are everywhere! They appear in electrical circuitry, mechanical systems, population ecology, roller coaster design, classrooms around the US, and even in the way Google’s search engine ranks pages.
MODULE 6 Learning Objectives
6.1: Factoring Out the Greatest Common Factor
- Factor out the Greatest Common Factor when it is a common monomial expression
- Factor out the Greatest Common Factor when it is a common parenthetical expression
6.2: Factoring by Grouping
- Factor a four-term polynomial by grouping
6.3: Factoring Trinomials with Leading Coefficient of 1
- Factor trinomials of the type [latex]x^2 + bx + c[/latex] by grouping
- Factor trinomials of the type [latex]x^2+bx+c[/latex] using the Product and Sum method
- Rewrite and factor trinomials of the above type by first factoring out the GCF
- Factor two-variable trinomials of the type [latex]x^2 + bxy + cy^2[/latex]
6.4: Factoring Trinomials with Leading Coefficient Other Than 1
- Factor trinomials of the type ax2 + bx + c, where a ≠ 1
- Factor trinomials of the above type where the GCF must first be factored out
6.5: Factoring Perfect Square Trinomials and the Difference of Squares
- Factor perfect square trinomials
- Factor the difference of two squares
- Factor difference of squares by first factoring out the GCF
6.6: Summary of Factoring
- Review the factoring methods presented in this module
- Be able to recognize and apply an appropriate factoring technique to a given problem
- Factor expressions completely
6.7: Solving Factorable Quadratic Equations
- Use factoring techniques and the Principle of Zero Products to solve polynomial equations
- Expand and then factor expressions to solve
6.8: Further Exploration with Quadratic Equations
- Solve application problems (area problems using factoring)
- Solve application problems (consecutive integer problems using factoring)
- Graph basic quadratic functions