This may sound a bit dumb, but is there any significant difference between factoring, grouping trinomials, difference of squares, and GCF?

Nothing significant, but there are important (however small) differences and they are used for different things.

How would you work out the problem if there were only 3 terms?

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!

Lets say there is no GCF in anything. Then What?

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.

what the meaning of GCF?

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.

Why are they called quadratics? They are typically trinomials with a leading term raised to the second degree.

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

so for your polynomials that you can not use the grouping method for what do you do to solve them ?

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

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?

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.

How does one factor by grouping when their is more than one letter in the equation. For example: 8ax+18a+20x+45

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

What is the point of this? I'm not going to use it when i do daily life things.

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!

Main content

Algebra 1

Course: Algebra 1 > Unit 13

Lesson 6: Factoring quadratics by grouping

Factoring by grouping

CCSS.Math:

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, 6, x, squared, plus, 4, x, equals, 2, x, left parenthesis, 3, x, plus, 2, right parenthesis .

What you will learn in this lesson

In this article, we will learn how to use a factoring method called grouping.

Example 1: Factoring 2, x, squared, plus, 8, x, plus, 3, x, plus, 12

First, notice that there is no factor common to all terms in 2, x, squared, plus, 8, x, plus, 3, x, plus, 12. 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:

start underbrace, left parenthesis, 2, x, squared, plus, 8, x, right parenthesis, end underbrace, start subscript, start text, f, i, r, s, t, space, g, r, o, u, p, i, n, g, end text, end subscript, plus, start underbrace, left parenthesis, 3, x, plus, 12, right parenthesis, end underbrace, start subscript, start text, s, e, c, o, n, d, space, g, r, o, u, p, i, n, g, end text, end subscript

In particular, there is a GCF of 2, x in the first grouping and a GCF of 3 in the second grouping. We can factor these out to obtain the following expression:

2, x, left parenthesis, x, plus, 4, right parenthesis, plus, 3, left parenthesis, x, plus, 4, right parenthesis

[How did you do that?]

Notice that this reveals yet another common factor between the two terms: start color #a75a05, x, plus, 4, end color #a75a05. We can use the distributive property to factor out this common factor.

2, x, left parenthesis, start color #a75a05, x, plus, 4, end color #a75a05, right parenthesis, plus, 3, left parenthesis, start color #a75a05, x, plus, 4, end color #a75a05, right parenthesis, equals, left parenthesis, start color #a75a05, x, plus, 4, end color #a75a05, right parenthesis, left parenthesis, 2, x, plus, 3, right parenthesis

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.

[I'd like to see this, please!]

Example 2: Factoring 3, x, squared, plus, 6, x, plus, 4, x, plus, 8

Let's summarize what was done above by factoring another polynomial.

\begin{aligned}&\phantom{=}3x^2+6x+4x+8\\\\ &=(3x^2+6x)+(4x+8)&&\small{\gray{\text{Group terms}}}\\ \\ &=3x({x+2})+4({x+2})&&\small{\gray{\text{Factor out GCFs}}}\\ \\ &=3x(\goldD{x+2})+4(\goldD{x+2})&&\small{\gray{\text{Common factor!}}}\\\\ &=(\goldD{x+2})(3x+4)&&\small{\gray{\text{Factor out } x+2}} \end{aligned}

The factored form is left parenthesis, x, plus, 2, right parenthesis, left parenthesis, 3, x, plus, 4, right parenthesis.

Check your understanding

1) Factor 9, x, squared, plus, 6, x, plus, 12, x, plus, 8.

(Choice A)
left parenthesis, 3, x, plus, 2, right parenthesis, left parenthesis, 3, x, plus, 4, right parenthesis
(Choice B)
left parenthesis, 3, x, plus, 1, right parenthesis, left parenthesis, 3, x, plus, 8, right parenthesis
(Choice C)
left parenthesis, 9, x, plus, 6, right parenthesis, left parenthesis, 12, x, plus, 8, right parenthesis
(Choice D)
left parenthesis, 3, x, plus, 2, right parenthesis, left parenthesis, 3, x, plus, 2, right parenthesis, left parenthesis, 3, x, plus, 4, right parenthesis

[I need help!]

2) Factor 5, x, squared, plus, 10, x, plus, 2, x, plus, 4.

[I need help!]

3) Factor 8, x, squared, plus, 6, x, plus, 4, x, plus, 3.

[I need help!]

Example 3: Factoring 3, x, squared, minus, 6, x, minus, 4, x, plus, 8

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 3, x, squared, minus, 6, x, minus, 4, x, plus, 8.

\begin{aligned}\phantom{0}&&&\phantom{=}3x^2-6x-4x+8\\\\ \small{\blueD{(1)}}&&&=(3x^2-6x)+(-4x+8)&&\small{\gray{\text{Group terms}}}\\\\ \small{\blueD{(2)}}&&&=3x(x-2)+(-4)(x-2)&&\small{\gray{\text{Factor out GCFs}}}\\\\ \small{\blueD{(3)}}&&&=3x(x-2)-4(x-2)&&\small{\gray{\text{Simplify}}}\\\\ \small{\blueD{(4)}}&&&=3x(\goldD{x-2})-4(\goldD{x-2})&&\small{\gray{\text{Common factor!}}}\\\\ \small{\blueD{(5)}}&&&=(\goldD{x-2})(3x-4)&&\small{\gray{\text{Factor out $x-2$}}}\\\\ \end{aligned}

The factored form of the polynomial is left parenthesis, x, minus, 2, right parenthesis, left parenthesis, 3, x, minus, 4, right parenthesis. We can multiply the binomials to check our work.

[I'd like to see this, please!]

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 start color #11accd, left parenthesis, 1, right parenthesis, end color #11accd, a "+" sign was added between the groupings left parenthesis, 3, x, squared, minus, 6, x, right parenthesis and left parenthesis, minus, 4, x, plus, 8, right parenthesis. This is because the third term left parenthesis, minus, 4, x, right parenthesis 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 3, x, squared, minus, 6, x, minus, 4, x, plus, 8 as left parenthesis, 3, x, squared, minus, 6, x, right parenthesis, minus, left parenthesis, 4, x, plus, 8, right parenthesis. This grouping, however, simplifies to 3, x, squared, minus, 6, x, minus, 4, x, start color #ca337c, minus, 8, end color #ca337c, which is not the same as the original expression.

[Can I see this?]

Why factor out minus, 4 instead of 4?

In step start color #11accd, left parenthesis, 2, right parenthesis, end color #11accd, we factored out a minus, 4 to reveal a common factor of left parenthesis, x, minus, 2, right parenthesis between the terms. If we instead factored out a positive 4, we would not obtain that common binomial factor seen above:

$\begin{aligned}(3x^2-6x)+(-4x+8)&=3x(\goldD{x-2})+4(\purpleC{-x+2})\\ \end{aligned}$

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 2, x, squared, minus, 3, x, minus, 4, x, plus, 6.

(Choice A)
left parenthesis, 2, x, plus, 3, right parenthesis, left parenthesis, x, minus, 2, right parenthesis
(Choice B)
left parenthesis, 2, x, minus, 3, right parenthesis, left parenthesis, x, minus, 2, right parenthesis
(Choice C)
left parenthesis, 2, x, plus, 3, right parenthesis, left parenthesis, 2, x, minus, 3, right parenthesis, left parenthesis, x, minus, 2, right parenthesis

[I need help!]

5) Factor 3, x, squared, plus, 3, x, minus, 10, x, minus, 10.

[I need help!]

6) Factor 3, x, squared, plus, 6, x, minus, x, minus, 2.

[I need help!]

Challenge problem

7*) Factor 2, x, cubed, plus, 10, x, squared, plus, 3, x, plus, 15.

[I need help!]

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 3, x, squared, plus, 9, x, plus, 2, x, plus, 6 since it can be written as follows:

$\begin{aligned}(3x^2+9x)+(2x+6)&=3x(\goldD{x+3})+2(\goldD{x+3})\\ \end{aligned}$

We cannot, however, use the grouping method to factor 2, x, squared, plus, 3, x, plus, 4, x, plus, 12 because factoring out the GCF from both groupings does not yield a common factor!

$\begin{aligned}(2x^2+3x)+(4x+12)&=x(\goldD{2x+3})+4(\purpleC{x+3})\\ \end{aligned}$

Using grouping to factor trinomials

You can also use grouping to factor certain three termed quadratics (i.e. trinomials) like 2, x, squared, plus, 7, x, plus, 3. This is because we can rewrite the expression as follows:

2, x, squared, plus, start color #11accd, 7, end color #11accd, x, plus, 3, equals, 2, x, squared, plus, start color #11accd, 1, end color #11accd, x, plus, start color #11accd, 6, end color #11accd, x, plus, 3

Then we can use grouping to factor 2, x, squared, plus, start color #11accd, 1, end color #11accd, x, plus, start color #11accd, 6, end color #11accd, x, plus, 3 as left parenthesis, x, plus, 3, right parenthesis, left parenthesis, 2, x, plus, 1, right parenthesis.

For more on factoring quadratic trinomials like these using the grouping method, check out our next article.

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Nin
Posted 6 years ago. Direct link to Nin's post “This may sound a bit dumb...”
This may sound a bit dumb, but is there any significant difference between factoring, grouping trinomials, difference of squares, and GCF?
Button navigates to signup pageComment on Nin's post “This may sound a bit dumb...”
(35 votes)
Answer
- Copperhead514
  Posted 3 years ago. Direct link to Copperhead514's post “Nothing significant, but ...”
  Nothing significant, but there are important (however small) differences and they are used for different things.
  Button navigates to signup page
  (23 votes)
daleighellison
Posted 6 years ago. Direct link to daleighellison's post “How would you work out th...”
How would you work out the problem if there were only 3 terms?
Button navigates to signup pageComment on daleighellison's post “How would you work out th...”
(14 votes)
Answer
- Ian Pulizzotto
  Posted 6 years ago. Direct link to Ian Pulizzotto's post “Find two numbers that add...”
  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!
  Comment on Ian Pulizzotto's post “Find two numbers that add...”
  (70 votes)
rama
Posted 5 years ago. Direct link to rama's post “what the meaning of GCF?”
what the meaning of GCF?
Button navigates to signup pageComment on rama's post “what the meaning of GCF?”
(2 votes)
Answer
- Kim Seidel
  Posted 5 years ago. Direct link to Kim Seidel's post “GCF is the abbreviation f...”
  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.
  Button navigates to signup page
  (21 votes)
Nikkia Metsker
Posted 3 years ago. Direct link to Nikkia Metsker's post “What is the point of this...”
What is the point of this? I'm not going to use it when i do daily life things.
Button navigates to signup pageComment on Nikkia Metsker's post “What is the point of this...”
(0 votes)
Answer
- J E
  Posted 3 years ago. Direct link to J E's post “You might be surprised! I...”
  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!
  Comment on J E's post “You might be surprised! I...”
  (25 votes)
Hafsa Mohamed
Posted 6 years ago. Direct link to Hafsa Mohamed's post “In problem 3, I solved th...”
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?
Button navigates to signup pageComment on Hafsa Mohamed's post “In problem 3, I solved th...”
(6 votes)
Answer
- assaagssadhchcgc
  Posted 2 months ago. Direct link to assaagssadhchcgc's post “u used _1.5_ which is not...”
  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)
  Button navigates to signup page
  (1 vote)
louisaandgreta
Posted 2 years ago. Direct link to louisaandgreta's post “Why are they called quadr...”
Why are they called quadratics? They are typically trinomials with a leading term raised to the second degree.
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- lemonomadic
  Posted 2 years ago. Direct link to lemonomadic's post “Quadratics actually are d...”
  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
  Comment on lemonomadic's post “Quadratics actually are d...”
  (10 votes)
Dannon Foster
Posted 5 months ago. Direct link to Dannon Foster's post “Lets say there is no GCF ...”
Lets say there is no GCF in anything. Then What?
Button navigates to signup pageComment on Dannon Foster's post “Lets say there is no GCF ...”
(3 votes)
Answer
- AuroraAlberts
  Posted a month ago. Direct link to AuroraAlberts's post “There is always a GCF, it...”
  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.
  Button navigates to signup page
  (6 votes)
Altangerel Chinzorig
Posted 9 months ago. Direct link to Altangerel Chinzorig's post “Why does it say we cannot...”
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?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- Kim Seidel
  Posted 9 months ago. Direct link to Kim Seidel's post “Your factors actually don...”
  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.
  Comment on Kim Seidel's post “Your factors actually don...”
  (7 votes)
katelynn Rodriguez
Posted 4 years ago. Direct link to katelynn Rodriguez's post “so for your polynomials t...”
so for your polynomials that you can not use the grouping method for what do you do to solve them ?
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- David Lee
  Posted 4 years ago. Direct link to David Lee's post “There is a formula for th...”
  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
  Button navigates to signup page
  (3 votes)
Dan Madsen
Posted 6 years ago. Direct link to Dan Madsen's post “How does one factor by gr...”
How does one factor by grouping when their is more than one letter in the equation. For example: 8ax+18a+20x+45
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- David Severin
  Posted 6 years ago. Direct link to David Severin's post “well, if you factor the f...”
  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
  Button navigates to signup page
  (3 votes)

Algebra 1

Course: Algebra 1 > Unit 13

What you need to know for this lesson

What you will learn in this lesson

Example 1: Factoring 2x2+8x+3x+122x^2+8x+3x+122x2+8x+3x+122, x, squared, plus, 8, x, plus, 3, x, plus, 12

Example 2: Factoring 3x2+6x+4x+83x^2+6x+4x+83x2+6x+4x+83, x, squared, plus, 6, x, plus, 4, x, plus, 8

Check your understanding

Example 3: Factoring 3x2−6x−4x+83x^2-6x-4x+83x2−6x−4x+83, x, squared, minus, 6, x, minus, 4, x, plus, 8

Check your understanding

Challenge problem

When can we use the grouping method?

Using grouping to factor trinomials

Want to join the conversation?

Example 1: Factoring 2, x, squared, plus, 8, x, plus, 3, x, plus, 12

Example 2: Factoring 3, x, squared, plus, 6, x, plus, 4, x, plus, 8

Example 3: Factoring 3, x, squared, minus, 6, x, minus, 4, x, plus, 8