Factoring polynomials involves expressing a polynomial as a product of simpler polynomials, known as factors. The factors of a polynomial are the polynomials that, when multiplied together, produce the original polynomial.
Key Facts
- Identify the polynomial: Determine the polynomial expression that you want to factor.
- Look for a greatest common factor (GCF): Check if there is a common factor that can be factored out from all the terms of the polynomial. This step helps simplify the polynomial and make factoring easier.
- Factor by grouping: If the polynomial has four or more terms, you can try grouping the terms into pairs and factor out a common factor from each pair. This method can help simplify the polynomial and make factoring more manageable.
- Use special factoring formulas: Look for special factoring formulas such as difference of squares, difference of cubes, or sum of cubes. These formulas can help you factor specific types of polynomials.
- Apply factoring techniques: Use factoring techniques such as the FOIL method (for trinomials) or trial and error to factor the polynomial further. These techniques involve finding two binomials that, when multiplied together, give you the original polynomial.
- Check for prime polynomials: If the polynomial cannot be factored any further, it is considered prime. Prime polynomials have no factors other than 1 and itself.
- Solve the equation: Once the polynomial is fully factored, you can use the zero product property to solve the equation by setting each factor equal to zero and solving for the variable.
It’s important to note that factoring polynomials can sometimes be complex and may require different techniques depending on the specific polynomial. It’s recommended to practice factoring with various examples to improve your skills.
Steps to Factor Polynomials
Identify the Polynomial
Determine the polynomial expression that you want to factor.
Look for a Greatest Common Factor (GCF)
Check if there is a common factor that can be factored out from all the terms of the polynomial. This step helps simplify the polynomial and make factoring easier.
Factor by Grouping
If the polynomial has four or more terms, you can try grouping the terms into pairs and factor out a common factor from each pair. This method can help simplify the polynomial and make factoring more manageable.
Use Special Factoring Formulas
Look for special factoring formulas such as difference of squares, difference of cubes, or sum of cubes. These formulas can help you factor specific types of polynomials.
Apply Factoring Techniques
Use factoring techniques such as the FOIL method (for trinomials) or trial and error to factor the polynomial further. These techniques involve finding two binomials that, when multiplied together, give you the original polynomial.
Check for Prime Polynomials
If the polynomial cannot be factored any further, it is considered prime. Prime polynomials have no factors other than 1 and itself.
Solve the Equation
Once the polynomial is fully factored, you can use the zero product property to solve the equation by setting each factor equal to zero and solving for the variable.
Example
Factor the polynomial: x² – 5x + 6
Step 1: Identify the Polynomial
The polynomial is x² – 5x + 6.
Step 2: Look for a GCF
The GCF of the terms x², -5x, and 6 is 1.
Step 3: Factor by Grouping
(x² – 5x) + 6
x(x – 5) + 6
Step 4: Use Special Factoring Formulas
Difference of squares: a² – b² = (a + b)(a – b)
(x – 2)(x – 3)
Step 5: Check for Prime Polynomials
The polynomial is now prime.
Step 6: Solve the Equation
(x – 2)(x – 3) = 0
x – 2 = 0 or x – 3 = 0
x = 2 or x = 3
Sources
FAQs
What is factoring a polynomial?
Factoring a polynomial is expressing it as a product of simpler polynomials, known as factors.
What is the first step in factoring a polynomial?
The first step is to identify the polynomial and look for a greatest common factor (GCF) that can be factored out from all the terms.
What if the polynomial has four or more terms?
If the polynomial has four or more terms, you can try grouping the terms into pairs and factoring out a common factor from each pair.
Are there any special formulas for factoring polynomials?
Yes, there are special factoring formulas for specific types of polynomials, such as the difference of squares, difference of cubes, and sum of cubes.
What if the polynomial cannot be factored any further?
If the polynomial cannot be factored any further, it is considered prime.
How can I check if my factored polynomial is correct?
You can check by multiplying the factors together and seeing if you get the original polynomial.
What is the zero product property?
The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero.
How can I use the zero product property to solve an equation?
You can set each factor equal to zero and solve for the variable.