5 Ways To Write A Polynomial In Factored Form

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When dealing with polynomials, expressing them in factored form can be incredibly useful for a variety of mathematical operations, such as solving equations, finding roots, and simplifying expressions. However, factoring polynomials can be a daunting task, especially for those who are new to algebra. Fortunately, there are several techniques that can help you write a polynomial in factored form.

In this article, we will explore five ways to write a polynomial in factored form. We will cover the basics of factoring, including the greatest common factor (GCF), difference of squares, sum and difference of cubes, and grouping. By the end of this article, you will have a solid understanding of how to factor polynomials and express them in factored form.

Method 1: Finding the Greatest Common Factor (GCF)

One of the simplest ways to factor a polynomial is to find the greatest common factor (GCF) of the terms. The GCF is the largest factor that divides all the terms of the polynomial without leaving a remainder.

For example, consider the polynomial 6x^2 + 12x + 18. To find the GCF, we can list the factors of each term:

• 6x^2: 1, 2, 3, 6
• 12x: 1, 2, 3, 4, 6, 12
• 18: 1, 2, 3, 6, 9, 18

The GCF of these terms is 6. Therefore, we can factor out 6 from each term:

6x^2 + 12x + 18 = 6(x^2 + 2x + 3)

Method 2: Difference of Squares

Another common factoring technique is the difference of squares. This technique involves factoring a polynomial that can be written in the form a^2 - b^2.

For example, consider the polynomial x^2 - 4. We can rewrite this polynomial as (x)^2 - (2)^2, which is a difference of squares. Therefore, we can factor it as:

x^2 - 4 = (x - 2)(x + 2)

Method 3: Sum and Difference of Cubes

The sum and difference of cubes are two other common factoring techniques. The sum of cubes involves factoring a polynomial that can be written in the form a^3 + b^3, while the difference of cubes involves factoring a polynomial that can be written in the form a^3 - b^3.

For example, consider the polynomial x^3 + 8. We can rewrite this polynomial as (x)^3 + (2)^3, which is a sum of cubes. Therefore, we can factor it as:

x^3 + 8 = (x + 2)(x^2 - 2x + 4)

Method 4: Grouping

Grouping is a factoring technique that involves grouping terms that have common factors. This technique is useful when factoring polynomials that have multiple terms.

For example, consider the polynomial x^2 + 3x + 2x + 6. We can group the terms as (x^2 + 3x) + (2x + 6). We can then factor out the GCF of each group:

(x^2 + 3x) + (2x + 6) = x(x + 3) + 2(x + 3)

We can then factor out the common binomial factor (x + 3):

x(x + 3) + 2(x + 3) = (x + 2)(x + 3)

Method 5: Using Synthetic Division

Synthetic division is a factoring technique that involves dividing a polynomial by a linear factor. This technique is useful when factoring polynomials that have a known root.

For example, consider the polynomial x^2 + 5x + 6. We can use synthetic division to divide this polynomial by the linear factor (x + 2). If the remainder is 0, then (x + 2) is a factor of the polynomial.

In conclusion, factoring polynomials is an essential skill in algebra. By using the five methods outlined above, you can write a polynomial in factored form and simplify expressions, solve equations, and find roots.