How to Factor by Grouping
Look at the equation., Find the master product., Separate the master product into its factor pairs., Find a factor pair with a sum equal to b., Split the center term into the two factors., Group the terms to form pairs., Factor out each pair...
Step-by-Step Guide
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Step 1: Look at the equation.
If you plan to use this method, the equation should follow a basic format of: ax2 + bx + c This process is usually used when the leading coefficient (the a term) is a number other than "1," but it can also be used for quadratic equations in which a =
1.
Example: 2x2 + 9x + 10 -
Step 2: Find the master product.
Multiply the a term and c term together.
The product of these two terms is referred to as the master product.
Example: 2x2 + 9x + 10 a = 2; c = 10 a * c = 2 * 10 = 20 , List the factors of your master product, separating them into their natural pairs (the pairs required to produced the master product).
Example:
The factors of 20 are: 1, 2, 4, 5, 10, 20 Written in factor pairs: (1, 20), (2, 10), (4, 5) , Look through the factor pairs and determine which set will produce the b term—the middle term and the coefficient of x—when added together.
If your master product was negative, you will need to find a pair of factors that equal the b term when subtracted from one another.
Example: 2x2 + 9x + 10 b = 9 1 + 20 = 21; this is not the correct pair 2 + 10 = 12; this is not the correct pair 4 + 5 = 9; this is the correct pair , Rewrite the center term, breaking it apart into the factor pair previously identified.
Make sure that you include the proper signs (plus or minus).
Note that the order of the center terms should not matter for this problem.
No matter which order you write the terms in, the end result should be the same.
Example: 2x2 + 9x + 10 = 2x2 + 5x + 4x + 10 , Group the first two terms into a pair and the second two terms into a pair.
Example: 2x2 + 5x + 4x + 10 = (2x2 + 5x) + (4x + 10) , Find the common factors of the pair and factor them out.
Rewrite the equation accordingly.
Example: x(2x + 5) + 2(2x + 5) , There should be a shared binomial parentheses between the two halves.
Factor this out, and place the other terms in another parentheses.
Example: (2x + 5)(x + 2) , You should now have your final answer.
Example: 2x2 + 9x + 10 = (2x + 5)(x + 2) The final answer is: (2x + 5)(x + 2) ,, -
Step 3: Separate the master product into its factor pairs.
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Step 4: Find a factor pair with a sum equal to b.
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Step 5: Split the center term into the two factors.
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Step 6: Group the terms to form pairs.
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Step 7: Factor out each pair.
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Step 8: Factor out shared parentheses.
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Step 9: Write your answer.
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Step 10: Factor: 4x2 - 3x - 10 a * c = 4 * -10 = -40 Factors of 40: (1
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Step 11: 8) Correct factor pair: (5
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Step 12: 8); 5 - 8 = -3 4x2 - 8x + 5x - 10 (4x2 - 8x) + (5x - 10) 4x(x - 2) + 5(x - 2) (x - 2)(4x + 5)
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Step 13: Factor: 8x2 + 2x - 3 a * c = 8 * -3 = -24 Factors of 24: (1
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Step 14: 6) Correct factor pair: (4
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Step 15: 6); 6 - 4 = 2 8x2 + 6x - 4x - 3 (8x2 + 6x) - (4x + 3) 2x(4x + 3) - 1(4x + 3) (4x + 3)(2x - 1)
Detailed Guide
If you plan to use this method, the equation should follow a basic format of: ax2 + bx + c This process is usually used when the leading coefficient (the a term) is a number other than "1," but it can also be used for quadratic equations in which a =
1.
Example: 2x2 + 9x + 10
Multiply the a term and c term together.
The product of these two terms is referred to as the master product.
Example: 2x2 + 9x + 10 a = 2; c = 10 a * c = 2 * 10 = 20 , List the factors of your master product, separating them into their natural pairs (the pairs required to produced the master product).
Example:
The factors of 20 are: 1, 2, 4, 5, 10, 20 Written in factor pairs: (1, 20), (2, 10), (4, 5) , Look through the factor pairs and determine which set will produce the b term—the middle term and the coefficient of x—when added together.
If your master product was negative, you will need to find a pair of factors that equal the b term when subtracted from one another.
Example: 2x2 + 9x + 10 b = 9 1 + 20 = 21; this is not the correct pair 2 + 10 = 12; this is not the correct pair 4 + 5 = 9; this is the correct pair , Rewrite the center term, breaking it apart into the factor pair previously identified.
Make sure that you include the proper signs (plus or minus).
Note that the order of the center terms should not matter for this problem.
No matter which order you write the terms in, the end result should be the same.
Example: 2x2 + 9x + 10 = 2x2 + 5x + 4x + 10 , Group the first two terms into a pair and the second two terms into a pair.
Example: 2x2 + 5x + 4x + 10 = (2x2 + 5x) + (4x + 10) , Find the common factors of the pair and factor them out.
Rewrite the equation accordingly.
Example: x(2x + 5) + 2(2x + 5) , There should be a shared binomial parentheses between the two halves.
Factor this out, and place the other terms in another parentheses.
Example: (2x + 5)(x + 2) , You should now have your final answer.
Example: 2x2 + 9x + 10 = (2x + 5)(x + 2) The final answer is: (2x + 5)(x + 2) ,,
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William Griffin
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