Factoring trinomials with two variables is a common task in algebra. A trinomial is a polynomial with three terms, and a two-variable trinomial is a trinomial that contains two variables. There are several methods for factoring two-variable trinomials, depending on the specific form of the trinomial.

### Key Facts

- Method for factoring trinomials with two variables:
- Step 1: Find the product and sum of the terms in the trinomial.
- Step 2: Look for two numbers that multiply to give the product and add up to give the sum.
- Step 3: Split the middle term of the trinomial using the two numbers found in Step 2.
- Step 4: Group the terms in pairs and factor out the greatest common factor (GCF) from each group.
- Step 5: Write the trinomial in factored form by combining the factored groups.

- Factoring trinomials with two variables and a leading coefficient of 1:
- For trinomials with a leading coefficient of 1, you can use the trial and error method or the grouping method to factor them.
- Example: e^2 – 6ef + 9f^2 can be factored as (e – 3f)^2.

- Factoring trinomials with two variables and a leading coefficient greater than 1:
- When the leading coefficient is greater than 1, you can still use the same methods mentioned above, but you may need to factor out the GCF first.
- Example: 18m^2 – 9mn – 2n^2 can be factored as (6m + n)(3m – 2n).

- Factoring trinomials with two variables using the AC method:
- The AC method involves finding two numbers that multiply to give the product of the leading coefficient and the constant term, and add up to give the coefficient of the middle term.
- Example: 30x^3y – 25x^2y^2 – 30xy^3 can be factored as 5xy(6x^2 – 5xy – 6y^2).

### Method for Factoring Trinomials with Two Variables

The general method for factoring trinomials with two variables is as follows:

- Find the product and sum of the terms in the trinomial.
- Look for two numbers that multiply to give the product and add up to give the sum.
- Split the middle term of the trinomial using the two numbers found in Step 2.
- Group the terms in pairs and factor out the greatest common factor (GCF) from each group.
- Write the trinomial in factored form by combining the factored groups.

### Factoring Trinomials with Two Variables and a Leading Coefficient of 1

For trinomials with a leading coefficient of 1, you can use the trial and error method or the grouping method to factor them.

## Trial and Error Method

The trial and error method involves trying different pairs of numbers until you find two numbers that work. For example, to factor the trinomial

e2−6ef+9f2e^2 – 6ef + 9f^2

, you would try different pairs of numbers that multiply to 9 and add up to -6. The pair of numbers that works is -3 and -3, so the trinomial can be factored as

(e−3f)2(e – 3f)^2

.

## Grouping Method

The grouping method involves grouping the first two terms and the last two terms of the trinomial, and then factoring out the GCF from each group. For example, to factor the trinomial

e2−6ef+9f2e^2 – 6ef + 9f^2

, you would group the first two terms as

e2−6efe^2 – 6ef

and the last two terms as

9f29f^2

. The GCF of the first group is

ee

, and the GCF of the second group is

9f9f

. So, the trinomial can be factored as

(e−3f)(e−3f)(e – 3f)(e – 3f)

.

### Factoring Trinomials with Two Variables and a Leading Coefficient Greater Than 1

When the leading coefficient is greater than 1, you can still use the same methods mentioned above, but you may need to factor out the GCF first.

## Example

To factor the trinomial

18m2−9mn−2n218m^2 – 9mn – 2n^2

, you would first factor out the GCF, which is 3. Then, you can use the grouping method to factor the remaining trinomial:

```
18m^2 - 9mn - 2n^2
= 3(6m^2 - 3mn - 2n^2)
= 3(2m - n)(3m + 2n)
```

### Factoring Trinomials with Two Variables Using the AC Method

The AC method involves finding two numbers that multiply to give the product of the leading coefficient and the constant term, and add up to give the coefficient of the middle term.

## Example

To factor the trinomial

30x3y−25x2y2−30xy330x^3y – 25x^2y^2 – 30xy^3

, you would first find the product of the leading coefficient and the constant term:

30×(−30)=−90030 \times (-30) = -900

. Then, you would find two numbers that multiply to -900 and add up to -25. The two numbers that work are -15 and 60. So, the trinomial can be factored as:

```
30x^3y - 25x^2y^2 - 30xy^3
= 5xy(6x^2 - 5xy - 6y^2)
```

### Conclusion

Factoring two-variable trinomials is a skill that can be used to solve a variety of algebraic problems. By understanding the different methods for factoring trinomials, you can quickly and easily factor any trinomial that you encounter.

### References

- Factoring Trinomials with Two Variables
- Factoring Polynomials with Four Terms and Two Variables
- [How to Factor Trinomials Lesson](https://www.greenemath.com/Algebra2/42/Howto

## FAQs

### What is a two-variable trinomial?

A two-variable trinomial is a polynomial with three terms, each of which contains two variables.

### How do you factor a two-variable trinomial?

There are several methods for factoring two-variable trinomials, including the trial and error method, the grouping method, and the AC method.

### What is the trial and error method?

The trial and error method involves trying different pairs of numbers until you find two numbers that work.

### What is the grouping method?

The grouping method involves grouping the first two terms and the last two terms of the trinomial, and then factoring out the GCF from each group.

### What is the AC method?

The AC method involves finding two numbers that multiply to give the product of the leading coefficient and the constant term, and add up to give the coefficient of the middle term.

### How do you know which method to use?

The best method to use for factoring a two-variable trinomial depends on the specific form of the trinomial.

### What are some examples of factoring two-variable trinomials?

x2−6xy+9y2=(x−3y)2x^2 – 6xy + 9y^2 = (x – 3y)^2

2×2+7xy−15y2=(2x−3y)(x+5y)2x^2 + 7xy – 15y^2 = (2x – 3y)(x + 5y)

30x3y−25x2y2−30xy3=5xy(6×2−5xy−6y2)30x^3y – 25x^2y^2 – 30xy^3 = 5xy(6x^2 – 5xy – 6y^2)