FACTORING POLYNOMIALS

Polynomials are foundational in mathematics, serving as the building blocks of algebraic expressions. Whether dealing with basic quadratic equations or intricate algebraic systems, polynomials are essential for describing numerous mathematical ideas. A key skill in working with these expressions is factoring, which involves breaking down a polynomial into a product of simpler polynomials. Like finding prime factors of numbers, factoring algebraic terms allows for more efficient equation solving, expression simplification, and a better understanding of algebraic relationships. Just as fractions represent parts of a whole, factoring decomposes complex expressions into easier-to-handle components, a crucial ability for solving equations, inequalities, and tackling integral calculus.

This blog post will discuss various methods for factoring polynomials and provide a detailed tutorial to help you become proficient in this lesson. An essential tool in your mathematical toolbox is knowing how to factor polynomials, regardless of whether you're a student trying to improve in math or someone interested in the real-world uses of algebra. 

1. Factoring by Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest factor that divides all terms in the polynomial. 

Steps:

1. Find the GCF of the numerical coefficients and the GCF of the variables by prime factorization. 
2. Factor out the GCF of the polynomial.
3. Rewrite the expression in factored form. 

Example:

If the leading coefficient is negative, always factor out the negative sign!

Examples

2. Factoring by Grouping

This method applies in situations with four terms. After grouping the terms together, factor in each pair of terms and then factor out the common binomial factor.

Steps:

3. Factoring Special Cases: Those are the special products of polynomials.

Let's recall those and then follow the pattern to find the factor.

A. Difference of Two Squares

a² - b² = (a + b)(a - b)

Steps:

1. Find the perfect squares of the first and second terms.

2. Separate the two factors of each term by writing the sum and difference of two similar terms

Examples:

B. Perfect Square Trinomial

a² + 2ab + b² = (a + b)(a + b) or (a + b)²

a² - 2ab + b² = (a - b)(a - b) or (a - b)²

Steps:

1. Find the perfect squares of the first and last terms.

2. Multiply 2 and the product of the two factors of the first and last terms to check if the middle term is correct.

3. Write it in factored form. Follow the pattern of factoring perfect square trinomials.

Examples:

C. Sum and Difference of Two Cubes

a² + b² = (a + b)(a² - ab + b²)

a² - b² = (a + b)(a² + ab + b²)

Steps:

1. Find the perfect cubes of the first and last terms.

2. For the remaining factors, square a, then multiply a and b, and lastly, square b.

3. Write it in factored form. Follow the pattern of factoring the sum and difference of two cubes.

Examples:

4. Factoring by General Trinomial

Recall the F.O.I.L. method when finding the product of two binomials.

A. Trinomials of the form x² + Bx + C

Case 1: When B and C are both positive, then the two factors of C are positive. 

Combining two positive integers through either multiplication or addition will yield a positive outcome.

Steps:

1. List all the possible factors whose product is C and whose sum is B.

2. Find the two factors.

3. Factor completely.

Example:

Case 2: When B is negative and C is positive, then the two factors of C are both negative. 

Multiplying two negative integers yields a positive result, while their sum is negative.

Steps:

1. List all the possible factors whose product is C and whose sum is B.

2. Find the two factors.

3. Factor completely.

Example:

Case 3: When B is positive and C is negative, then the two factors of C are of opposite signs, but the higher number will be a positive sign. 

Multiplying a positive and a negative integer yields a negative result. For the addition of opposite signs, the sign of the larger absolute value prevails, and the integers are subtracted.

Steps:

1. List all the possible factors whose product is C and whose sum is B.

2. Find the two factors.

3. Factor completely.

Example:

Case 3: When B and C are both negative, then the two factors of C are of opposite signs, but the higher number will be a negative sign. 

A negative result is obtained when a positive and a negative integer are multiplied. When adding integers with different signs, you subtract the numbers after taking the sign of the number with the higher absolute value.

Steps:

1. List all the possible factors whose product is C and whose sum is B.

2. Find the two factors.

3. Factor completely.

Example:

B. Trinomials of the form Ax² + Bx + C

Steps:

1. Identify the numbers of A and C, then find their product AC.

2. List all the possible factors whose product is AC and whose sum is B.

3. Use the factors to write Bx as the sum of two terms.

4. Factoring by grouping.

5. Factor completely.

Example:

If the terms share common factors, begin by finding the Greatest Common Factor (GCF). Once you've factored out the GCF, proceed with the standard steps for factoring a general trinomial.

Example:

Here's the videos!

Video 1: Factoring Polynomials by GCF or by Monomials

Video 2: Factoring Polynomials by Grouping