- Intro to grouping
- Factoring by grouping
- Factoring quadratics by grouping
- Factoring quadratics: leading coefficient ≠ 1
- Factor quadratics by grouping
- Factoring quadratics: common factor + grouping
- Factoring quadratics: negative common factor + grouping
- Creativity break: How can we combine ways of thinking in problem solving?
Factoring by grouping
Learn about a factorization method called "grouping." For example, we can use grouping to write 2x²+8x+3x+12 as (2x+3)(x+4).
What you need to know for this lesson
Factoring a polynomial involves writing it as a product of two or more polynomials. It reverses the process of polynomial multiplication.
We have seen several examples of factoring already. However, for this article, you should be especially familiar with taking common factors using the distributive property. For example, .
What you will learn in this lesson
In this article, we will learn how to use a factoring method called grouping.
Example 1: Factoring
First, notice that there is no factor common to all terms in . However, if we group the first two terms together and the last two terms together, each group has its own GCF, or greatest common factor:
In particular, there is a GCF of in the first grouping and a GCF of in the second grouping. We can factor these out to obtain the following expression:
Notice that this reveals yet another common factor between the two terms: . We can use the distributive property to factor out this common factor.
Since the polynomial is now expressed as a product of two binomials, it is in factored form. We can check our work by multiplying and comparing it to the original polynomial.
Example 2: Factoring
Let's summarize what was done above by factoring another polynomial.
The factored form is .
Check your understanding
1) Factor .
2) Factor .
3) Factor .
Example 3: Factoring
Extra care should be taken when using the grouping method to factor a polynomial with negative coefficients.
For example, the steps below can be used to factor .
The factored form of the polynomial is . We can multiply the binomials to check our work.
A few of the steps above may seem different than what you saw in the first example, so you may have a few questions.
Where did the "+" sign between the groupings come from?
In step , a "+" sign was added between the groupings and . This is because the third term is negative, and the sign of the term must be included within the grouping.
Keeping the minus sign outside the second grouping is tricky. For example, a common error is to group as . This grouping, however, simplifies to , which is not the same as the original expression.
Why factor out instead of ?
In step , we factored out a to reveal a common factor of between the terms. If we instead factored out a positive , we would not obtain that common binomial factor seen above:
When the leading term in a group is negative, we will often need to factor out a negative common factor.
Check your understanding
4) Factor .
5) Factor .
6) Factor .
7*) Factor .
When can we use the grouping method?
The grouping method can be used to factor polynomials whenever a common factor exists between the groupings.
For example, we can use the grouping method to factor since it can be written as follows:
We cannot, however, use the grouping method to factor because factoring out the GCF from both groupings does not yield a common factor!
Using grouping to factor trinomials
You can also use grouping to factor certain three termed quadratics (i.e. trinomials) like . This is because we can rewrite the expression as follows:
Then we can use grouping to factor as .
For more on factoring quadratic trinomials like these using the grouping method, check out our next article.
Want to join the conversation?
- This may sound a bit dumb, but is there any significant difference between factoring, grouping trinomials, difference of squares, and GCF?(35 votes)
- Nothing significant, but there are important (however small) differences and they are used for different things.(23 votes)
- How would you work out the problem if there were only 3 terms?(14 votes)
- Find two numbers that add to the middle coefficient and multiply to give the product of the first and last coefficients (or constants). This is called the ac method.
Example: Factor 6x^2 + 19x + 10.
6*10 = 60, so we need to find two numbers that add to 19 and multiply to give 60. These numbers (after some trial and error) are 15 and 4. So split up 19x into 15x + 4x (or 4x + 15x), then factor by grouping:
6x^2 + 19x + 10 = 6x^2 + 15x + 4x + 10
= 3x(2x + 5) + 2(2x + 5)
= (3x + 2)(2x + 5).
Have a blessed, wonderful day!(70 votes)
- what the meaning of GCF?(2 votes)
- GCF is the abbreviation for Greatest Common Factor.
It is the value that you can evenly divide all terms by. It can be a number, a variable, or a mix of numbers and variables.(21 votes)
- What is the point of this? I'm not going to use it when i do daily life things.(0 votes)
- You might be surprised! If you decide to go into a career like engineering or to work on cool projects like an autonomous car or a rocket, you will use polynomials on a daily basis to solve problems like how to control the car using different sensor inputs. Knowing how to manipulate polynomials like this is therefore extremely useful when designing these systems!(25 votes)
- In problem 3, I solved the expression and got the answer (4x+2)(2x+1.5). It said my answer was correct, but when I checked my answer, I got the same expression. What did I do wrong?(6 votes)
- u used 1.5 which is not a natural number. Therefore answer is incorrect. Correct answer must include only natural numbers: (3x+2)(3x+4)(1 vote)
- Why are they called quadratics? They are typically trinomials with a leading term raised to the second degree.(2 votes)
- Quadratics actually are derived from the Latin word, 'quadratum', which literally means 'square'.
A quadratic is basically a type of problem that deals with a variable multiplied by itself — an operation known as squaring.
So, you could relate the word quadratic to Latin, not mathematics(10 votes)
- Lets say there is no GCF in anything. Then What?(3 votes)
- There is always a GCF, it just depends on how many there are.
Take 8 and 6, the factors of those two numbers are 1 and 2.
2 is bigger, so the GCF of 8 and 6 is 2.
That works the same with every other set of numbers.
Let's take 4 and 3.
The factors of 3 are 1,3
The factors of 4 are 1,2,4
The only common factor here is 1, so 1 is the GCF of 4 and 3!
So to answer your question, if you can't find the greatest common factor of two numbers, it's 1.(6 votes)
- Why does it say we cannot use grouping on this one 2x^2+3x+4x+12. It is usable though isn't it? Answer would be (x+2)(2x+3) right?(1 vote)
- Your factors actually don't create the polynomial. If you multiply your factors, you will get: 2x^2+3x+4x+6, which does not match the original polynomial.
Now, how do you know if isn't working before checking the factors by multiplying them.
Pull the GCF from each pair of terms to get:
2x^2+3x+4x+12 = x(2x+3) + 4(x+3)
Notice, the 2 binomials in parentheses do not match. To go to factors from this point, we need to remove a common binomial factor from each term. There is no common binomial factor. This tells use that the polynomial is not factorable.
Hope this helps.(7 votes)
- so for your polynomials that you can not use the grouping method for what do you do to solve them ?(2 votes)
- There is a formula for that. Here is the link (for quadratic): https://www.khanacademy.org/math/algebra/quadratics/solving-quadratics-using-the-quadratic-formula/a/quadratic-formula-review(3 votes)
- How does one factor by grouping when their is more than one letter in the equation. For example: 8ax+18a+20x+45(1 vote)
- well, if you factor the first two to get 2a (4x + 9) and the second two as 5 (4x+9), you can see that an equivalent would be (2a + 5)(4x+9). Or if you factored first and third tern and second and fourth, you get 4x(2a+5) + 9(2a+5) and get the same answer(3 votes)