What are Polynomials?
Polynomials are expressions that contain variables and coefficients. They only use addition, subtraction, multiplication, and exponents that aren’t negative integers. You’ve probably seen them before. Some examples are x² + 2x + 1, 3x³ – 5x + 2, and even just the number 7.
Why is Factoring Important?
Factoring simplifies algebraic expressions, making them easier to work with. It’s a key skill for solving equations and understanding more advanced math. Basically, mastering factoring is essential for success as you move forward in math.
Factoring Methods
There are several methods for factoring, including Difference of Two Squares, Trinomial Factoring, Completing the Square, and Factoring by Grouping. Need practice? This page has a variety of factoring polynomials worksheets to help you hone your skills.
Factoring the Difference of Two Squares
One of the most common types of factoring you’ll learn in algebra class is factoring the difference of two squares. This technique is used for binomials, that is, polynomials with two terms.
Understanding the concept
The difference of two squares is an expression that takes the form a² – b². Factoring this type of expression results in (a + b)(a – b).
To use this technique, you’ll need to recognize perfect squares (1, 4, 9, 16, 25, 36, etc.) and a subtraction operation. For example, x² – 9 and 4x² – 25 both fit this pattern.
Factoring process with examples
Here’s a step-by-step guide to factoring the difference of two squares:
- Identify ‘a’ and ‘b’ in the expression.
- Apply the formula (a + b)(a – b).
For example, let’s factor x² – 25.
- a = x, b = 5
- Solution: (x + 5)(x – 5)
Here’s another example: 4x² – 9
- a = 2x, b = 3
- Solution: (2x + 3)(2x – 3)
Common mistakes to avoid
When you’re factoring the difference of two squares, watch out for these common mistakes:
- Incorrectly identifying perfect squares.
- Forgetting the plus/minus sign in the factored form.
- Trying to factor a sum of squares (a² + b²), which generally can’t be factored using real numbers.
Factoring Trinomials
Factoring trinomials can seem tricky, but with a little practice, you’ll start to recognize patterns that can make the process easier. Here’s how to factor trinomials in the form x² + bx + c and ax² + bx + c.
Factoring Trinomials of the Form x² + bx + c
In this type of trinomial, you’re looking for two numbers that add up to b and multiply to c. Once you find those numbers, you can write the factored form as (x + number1)(x + number2).
Let’s look at a couple of examples.
Example 1: Factoring x² + 2x – 15
We need two numbers that add up to 2 and multiply to -15. The numbers 5 and -3 fit the bill, since 5 + (-3) = 2 and 5 -3 = -15.
The solution is (x + 5)(x – 3).
Example 2: Factoring x² – 5x + 6
In this case, we need two numbers that add up to -5 and multiply to 6. The numbers -2 and -3 work because -2 + (-3) = -5 and -2 -3 = 6.
The solution is (x – 2)(x – 3).
Factoring Trinomials of the Form ax² + bx + c (where a ≠ 1)
When there’s a coefficient in front of the x² term, you can use the AC method. Multiply a and c, then find two numbers that multiply to ac and add up to b. Replace the bx term with the two numbers you found, and then factor by grouping.
Here’s an example:
Factoring 2x² + 7x + 3
First, multiply a and c: 2 3 = 6. Now, find two numbers that multiply to 6 and add up to 7. The numbers 6 and 1 work.
Next, replace the 7x term with 6x + x: 2x² + 6x + x + 3
Now, factor by grouping: 2x(x + 3) + 1(x + 3)
The solution is (2x + 1)(x + 3).
Tips for Factoring Trinomials
- Practice recognizing patterns and number combinations.
- Use trial and error, but do it systematically.
- Always check your work by expanding the factored form to make sure it matches the original trinomial.
Factoring by Grouping
Factoring by grouping is a good strategy when you have polynomials with four or more terms. The basic idea is to look for common factors within pairs of terms.
Here’s how it works:
- Group terms.
- Factor out the greatest common factor from each group.
- Factor out the common binomial.
Let’s look at a couple of examples:
Example 1: Factor x³ + 2x² + 3x + 6
- Group: (x³ + 2x²) + (3x + 6)
- Factor out the greatest common factor: x²(x + 2) + 3(x + 2)
- Factor out the common binomial: (x² + 3)(x + 2)
Solution: (x² + 3)(x + 2)
Example 2: Factor 2x³ – 3x² – 4x + 6
- Group: (2x³ – 3x²) + (-4x + 6)
- Factor out the greatest common factor: x²(2x – 3) – 2(2x – 3)
- Factor out the common binomial: (x² – 2)(2x – 3)
Solution: (x² – 2)(2x – 3)
Factoring by completing the square
Sometimes, you’ll run into a quadratic expression that’s difficult to factor using the usual methods. That’s when you may want to try “completing the square.” This technique transforms the quadratic expression into a perfect square trinomial plus a constant.
How to complete the square
- Make sure the coefficient of x² is 1. If it isn’t, divide the whole expression by that coefficient.
- Take half of the coefficient of the x term, square it, and then add and subtract it within the expression.
- Rewrite the expression as a perfect square trinomial plus or minus a constant.
Example: Factoring x² + 10x + 5
- Take half of 10 (which is 5), and square it (25). Add and subtract it from the expression.
- x² + 10x + 25 – 25 + 5
- (x + 5)² – 20
- Solution: (x + 5)² – 20 (or x = -5 ± 2√5, if you’re solving for x)
Why is completing the square so hard?
Completing the square involves a lot of steps that require careful attention to detail. It can be hard to remember the formulas and apply them correctly, so practice is essential.
In Summary
We’ve covered a lot of ground in this overview of factoring polynomials. Remember the key methods: spotting the Difference of Two Squares, mastering Trinomial Factoring, using Factoring by Grouping when you have more terms, and understanding how and when to Complete the Square.
The secret to getting good at factoring isn’t just understanding the rules, though. You need to practice, practice, practice! Working through problems on worksheets is one of the best ways to build your skills and confidence.
So, keep at it! If you get stuck, don’t be afraid to ask for help from your teacher or a tutor. There are also tons of great resources online and in textbooks that can give you extra practice and explanations. With a little effort, you’ll be factoring polynomials like a pro in no time!