Factoring Polynomials: Techniques and Examples

Factoring polynomials is a fundamental skill in algebra. It helps us simplify tricky math problems by breaking them into smaller, manageable parts.

In this article, we’ll discuss different ways to do factoring polynomials.

What is Factoring a Polynomial?

Factoring a polynomial means finding its factors, which are expressions that, when multiplied together, yield the original polynomial. It is the reverse process of multiplication.

Common Factoring Techniques

1: Greatest Common Factor (GCF)

Often, a polynomial has a common factor in all its terms. To factor this out, follow these steps:

• Identify the greatest common factor (GCF) of all the terms in the polynomial.
• Divide each term by the GCF.
• Express the polynomial as the product of the GCF and the simplified expression.

Example

Factoring the GCF of 6x² + 9x³:

Step 1: GCF of 6x² and 9x³ is 3x².

Step 2: Divide each term by 3x²:

6x²/3x² + 9x³/3x² = 2 + 3x

Step 3: The factored polynomial is 3x²(2 + 3x).

2: Factor by Grouping

Factor by grouping method works for polynomials with four terms. You group the first two terms and last two terms, factor each group, and then factor out the common factor:

Example

x² + 5x + 6x + 30

= (x²+ 5x) + (6x + 30)

= x(x + 5) + 6(x + 5)

= (x + 5)(x + 6)

3: Factor Trinomials

Trinomials are polynomials with three terms. To factor a trinomial in the form ax² + bx + c, follow these steps:

• Find factors of c that add up to b
• Split the middle term bx into the two factors found
• Factor each group

Example

x²+ 7x + 12

Factors of 12 that add to 7 are 3 and 4

Split 7x into 3x + 4x

= x² + 3x + 4x + 12

= x(x + 3) + 4(x + 3)

= (x + 3)(x + 4)

4: Difference of Squares

Difference of squares occurs when you have a binomial in the form a² – b². This can be factored as:

(a + b)(a – b)

Example

9x² – 4

9x² – 4 = (3x)² – 2²

Factor as difference of squares:

= (3x + 2)(3x – 2)

5: Factoring by Grouping Polynomials

Factor by grouping method can work for polynomials with more than 4 terms. You group terms together, factor each group, and factor out the common factor.

Example

= 6x³ + 3x² – 4x – 2

= (6x³ + 3x²) + (-4x – 2)

= 3x(2x² + x) – 2(2x + 1)

= 3x(2x + 1)(x – 2)

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