Factoring Polynomials - Algebra I | Luna Learn 24

Factoring Polynomials - Algebra I

Lesson Description: Factoring polynomials is a crucial skill in Algebra. This lesson will cover methods to factor different types of polynomials, including common factors, trinomials, and the difference of squares.

1. Introduction to Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into its component factors. For example, factoring \( x^2 - 5x + 6 \) results in \( (x - 2)(x - 3) \). Understanding how to factor will help in solving equations and simplifying expressions.

2. Methods of Factoring

Factoring out the Greatest Common Factor (GCF)
Factoring Trinomials
Factoring by Grouping
Factoring the Difference of Squares

3. Examples

Example 1: Factor \( 2x^3 + 4x^2 - 6x \).

Solution: The GCF is \( 2x \). Factoring out the GCF gives: \( 2x(x^2 + 2x - 3) \).

Example 2: Factor \( x^2 + 5x + 6 \).

Solution: The factors of 6 that add up to 5 are 2 and 3. So, \( x^2 + 5x + 6 = (x + 2)(x + 3) \).

Example 3: Factor \( 4x^2 - 9 \).

Solution: This is a difference of squares: \( 4x^2 - 9 = (2x - 3)(2x + 3) \).

Example 4: Factor \( x^3 + 3x^2 + 3x + 9 \) by grouping.

Solution: Grouping terms gives: \( (x^3 + 3x^2) + (3x + 9) \). Factoring each group: \( x^2(x + 3) + 3(x + 3) = (x^2 + 3)(x + 3) \).

Example 5: Factor \( 6x^2 + 11x - 10 \).

Solution: Using the method of grouping, we find: \( 6x^2 + 15x - 4x - 10 = 3x(2x + 5) - 2(2x + 5) = (3x - 2)(2x + 5) \).

Example 6: Factor \( x^2 - 16 \).

Solution: This is a difference of squares: \( x^2 - 16 = (x - 4)(x + 4) \).

4. Exercises

Try factoring the following polynomials:

Factor \( x^2 + 7x + 10 \).
Factor \( 3x^2 - 12x + 9 \).
Factor \( x^2 - 25 \).
Factor \( 2x^3 - 6x^2 + 4x \).
Factor \( x^3 - 2x^2 - x + 2 \) by grouping.

5. Solutions

Exercise 1 Solution: \( x^2 + 7x + 10 = (x + 2)(x + 5) \)

Exercise 2 Solution: \( 3x^2 - 12x + 9 = 3(x^2 - 4x + 3) = 3(x - 1)(x - 3) \)

Exercise 3 Solution: \( x^2 - 25 = (x - 5)(x + 5) \)

Exercise 4 Solution: \( 2x^3 - 6x^2 + 4x = 2x(x^2 - 3x + 2) = 2x(x - 1)(x - 2) \)

Exercise 5 Solution: \( x^3 - 2x^2 - x + 2 = (x^2 - 1)(x - 2) = (x - 1)(x + 1)(x - 2) \)

6. Homework

Complete the following problems and submit your answers:

Factor \( x^2 + 4x + 4 \).
Factor \( 5x^2 - 30x + 45 \).
Factor \( x^3 + 2x^2 - x - 2 \) by grouping.

7. Revision

Review the key methods for factoring polynomials, including factoring out the GCF, factoring trinomials, and factoring by grouping. Practice identifying different forms of polynomials and apply the appropriate factoring technique.