Unit 9: Factoring Polynomials
Factoring is multiplication in reverse: rather than multiplying two polynomials, you write a given polynomial as a product of two or more different expressions. Factoring is an important tool for solving advanced equations, such as quadratic equations. Quadratic equations occur in problems that involve rectangular objects and their areas, such as planning gardens, framing photographs, or carpeting a floor.
Completing this unit should take you approximately 7 hours.
9.1: Identifying Greatest Common Factor
Reading and understanding algebraic expressions, much like translating sentences from a foreign language, is a skill that takes time to develop. Before learning how to factor an algebraic expression (or write the expression as a product), you need to be able to break down the parts.
9.2: Factoring Polynomials by Grouping
There are four steps you need to take to factor a four-term polynomial:
- Separate the polynomial into two groups of two terms.
- Identify a common factor in each member of the group and factor it out.
- Check that the resulting expressions contain a common binomial factor. If they do not, then the polynomial cannot be factored (at least not when the terms are grouped this way).
- Factor out a common binomial factor, and rewrite the polynomial as a product of two binomials.
9.3: Revisiting FOIL by Working Backwards
When we factor binomials, we are essentially "undoing" the FOIL technique we learned in the last unit.
9.4: Factoring Trinomials of the Form x² + bx + c when c is Positive
We will study how to factor trinomials in two different forms: when c is positive and when c is negative. In this section, we will study how to factor trinomials with a positive c value.
9.5: Factoring Trinomials of the Form x² + bx + c when c is Negative
Now we can study trinomials where c is a negative number. These can be more complicated.
9.6: Factoring by Grouping
One useful method for factoring polynomials is factoring by grouping, using the bridge or the AC method.
9.7: Factoring Trinomials of the Form x² + bxy + cy² and ax² + bxy + cy²
The previous examples of factoring trinomials only included one variable. Often, algebraic expressions include more than one variable, and we need ways to factor polynomials with two variables. The methods used to factor trinomials with two variables are based on the same principles as those with one variable.
9.8: Identifying and Factoring Complete Square Trinomials
We can identify some trinomials as perfect squares. A perfect square trinomial has the form a2 + 2ab + b2 or a2 − 2ab + b2. Because each term is a square, they can be factored into (a + b)2 or (a − b)2, respectively. It is important to be able to identify trinomials in this form to easily factor them after determining the values of a and b.
9.9: Identifying and Factoring the Difference of Two Squares
A special case of polynomials is the difference of two squares. For example, the equation x2 − 4 is a difference of two squares because it can be written as x2 − 22.
9.10: Identifying and Factoring the Sum and Difference of Two Cubes
Much like factoring the difference of two squares, we can use special techniques to factor the sum and difference of two cubes. An example of the sum of two cubes is x3 + 27, as this can be written as x3 + 33.
9.11: Factoring General Polynomials
Sometimes factoring a polynomial requires more than one step. When this occurs, we combine the different techniques we have learned to complete factor the complex polynomial.
9.12: The Principle of Zero Product and Identifying Solutions
The principle of zero product tells us that if the product of two quantities is 0, one of the quantities must also be 0.
9.13: Using Factoring to Solve Quadratic Equations
We are now ready to use factoring to solve quadratic equations.
9.14: Applications for Solving Problems
Now that we have spent so much time learning about factoring, we are ready to see how it applies to everyday life.