Factoring – Explanation, Types, GCF, Problems, Rules, and FAQs

Introduction to Factoring

Factoring is the process of dividing a number into a product of prime numbers. The number is called the “factor” of the product, and the prime numbers are the “factors.”

For example, the factorization of 60 is 2 x 2 x 3 x 5. This means that 60 can be divided evenly into 2, 2, 3, 5, 10, 15, and 30.

Types of Factoring Algebraically

There are three types of factoring algebraically:

1) Trinomial factoring: This is the process of factoring a trinomial, which is an algebraic expression with three terms. The expression is factored into two binomial expressions, each of which is the product of a factor and a sum or difference of the two terms of the original trinomial.

2) Quadratic factoring: This is the process of factoring a quadratic equation, which is an algebraic equation with two terms. The equation is factored into two binomial expressions, each of which is the product of a factor and the square of the other term.

3) Polynomial factoring: This is the process of factoring a polynomial, which is an algebraic expression with more than three terms. The expression is factored into two or more binomial expressions, each of which is the product of a factor and a sum or difference of the terms in the original polynomial.

Greatest Common Factor

The greatest common factor (GCF) of two or more numbers is the largest number that evenly divides all of the numbers. The GCF is also the highest common factor (HCF).

For example, the GCF of 12 and 15 is 3 because 3 evenly divides both 12 and 15. The GCF of 15 and 30 is 5 because 5 evenly divides both 15 and 30.

Factorisation Problems:

1. Factorise the following into prime factors:

162

2. Factorise the following into prime factors:

24

3. Factorise the following into prime factors:

120

4. Factorise the following into prime factors:

288

5. Factorise the following into prime factors:

720

1. 162 = 2 x 2 x 3 x 3
2. 24 = 2 x 2 x 3
3. 120 = 2 x 2 x 3 x 5
4. 288 = 2 x 2 x 2 x 3 x 3
5. 720 = 2 x 2 x 2 x 3 x 3 x 5

Factoring Polynomial by Grouping

Factor the polynomial by grouping the terms.

x2 + 3x – 2

(x + 2)(x – 1)

Factoring Rules:

1. The first rule of factoring is that a number can be factored into its prime factors. For example, the number 12 can be factored into 2 × 2 × 3.

2. The second rule of factoring is that a number can be factored into the product of its prime factors. For example, the number 144 can be factored into 2 × 2 × 2 × 3 × 3.