Factorization Calculator
Free math practice app for middle school students
(Example)x^2+4x+3
How to Use the Calculator
Enter an expression
Type the expression you want to factorize into the input field. Example: x^2+4x+3
Use the keyboard for helper input
Use the on-screen keyboard with x/y/z, ^2, ^3, (), +, and - to build the expression.
Run the factorization
Click Factorize to display the factored form along with a short explanation.
Explanation of Factorization
Factorization means rewriting a polynomial as a product. It may look difficult at first, but it is simply the process of turning an expression into multiplied factors using parentheses.
Expansion removes parentheses, while factorization does the reverse and puts the expression back into a factored form.
You can check whether a factorization is correct by expanding the result and confirming that it returns to the original expression.
x^2 - 5x = x(x-5)
x^2 + 6x + 5 = (x+1)(x+5)
Basic: Factor Out the Common Factor
The first step in factorization is to look for a common factor.
When each term shares the same number or variable part, that shared part can be factored out.
xy + 8y
y is common, so = y(x+8)
x^2 + 4x
x is common, so = x(x+4)
5x^2 - 15x
5 and x are common, so = 5x(x-3)
Basic: Formula 1
Formula 1
x^2 + (a+b)x + ab = (x+a)(x+b)
The key is to find a and b so that their sum matches the middle term and their product matches the constant term.
x^2 + 6x + 5
6 = (1+5), 5 = (1×5)
= (x+1)(x+5)
When (a+b)x and (a×b) are positive:
x^2 + (a+b)x + ab = (x+a)(x+b)
When (a×b) is negative:
x^2 + (a+b)x - ab = (x-a)(x+b)
When (a+b)x is negative and (a×b) is positive:
x^2 - (a+b)x + ab = (x-a)(x-b)
Basic: Formula 2
Formula 2
x^2 ± 2ax + a^2 = (x ± a)^2
The key idea is that the middle term must be twice the square root of the last term.
x^2 + 6x + 9 = (x+3)^2
x^2 + 8x + 16 = (x+4)^2
x^2 - 8x + 16 = (x-4)^2
Basic: Formula 3
Formula 3
x^2 - a^2 = (x+a)(x-a)
x^2 - 16 = (x+4)(x-4)
-9 + x^2 → x^2 - 9 = (x+3)(x-3)
Basic: Cross Multiplication
Use cross multiplication by checking the product and the diagonal sum to factor the quadratic.
3x^2 - 11x + 6
→ Choose factors of 3 and 6 whose diagonal sum is -11
→ (3x-2)(x-3)
Application: Factor Out a Common Factor Then Use a Formula
4x^2y + 28xy + 48y
= 4y(x^2+7x+12)
= 4y(x+3)(x+4)
ax^2 - 10ax + 25a
= a(x^2-10x+25)
= a(x-5)^2
3x^3 - 48x
= 3x(x^2-16)
= 3x(x+4)(x-4)
Application: Substitution
x(a+b)-7(a+b)
Let (a+b) = A → xA-7A = A(x-7)
Substitute A back → (a+b)(x-7)
(a+b)^2-7(a+b)+10
Let (a+b) = A → A^2-7A+10
= (A-2)(A-5)
Substitute A back → (a+b-2)(a+b-5)
Application: Repeated Factorization
x^2+10x+25-36a^2
= (x+5)^2 - 36a^2
= (x+5+6a)(x+5-6a)
a^2 - x^2 + 6x - 9
= a^2 - (x-3)^2
= (a+x-3)(a-x+3)
* If a minus sign is in front of A, flip the signs inside
Application: Factor by Grouping
xa - xb - 5a + 5b
= (xa-xb) - (5a-5b)
= x(a-b) - 5(a-b)
= (a-b)(x-5)
* If a minus sign is outside parentheses, flip the signs inside
Example Factorization Problems
4x^2+16x
A:
Factor out the common factor 4x
= 4x(x+4)
y^2-7y-18
A:
Find two numbers whose sum is -7 and product is -18
= (y-9)(y+2)
a^2-24a+144
A:
Apply Formula 2
= (a-12)^2
4x^2y-36y
A:
Factor out 4y
= 4y(x^2-9)
Apply Formula 3
= 4y(x+3)(x-3)
Where You Might Use This
- 1Middle school or high school math homework and test preparation
- 2Checking expressions that you solved by hand
- 3Learning the relationship between expansion and factorization
- 4Preprocessing equations before graphing problems
- 5Simplifying expressions that appear in word problems
Examples of Invalid Input
The following expressions will cause an input error. Correct the expression and try again.
Input format errors
(blank)
The expression is empty, so it cannot be calculated.
Enter an expression with at least one character. Example: x^2+4x+3
x^2+1=0
An equals sign is not allowed.
Enter only the left-hand side. Example: x^2+1
x^^2+1
The exponent format is invalid.
Write only one integer after ^. Example: x^2+1
(x+1
The parentheses are not closed.
Every opening parenthesis must be closed. Example: (x+1)(x+2)
Currently unsupported input
x^-2+1
Negative exponents are not supported.
Rewrite the expression without negative exponents.
x/2+1
This calculator does not support this slash-based division input format.
Rewrite the expression with expanded integer coefficients before entering it.
Expressions outside the supported conditions
2+3
Expressions without variables are not supported.
Enter an expression that contains a variable. Example: x^2-5x
x^7+1
Currently only expressions up to degree 6 are supported.
Split the expression into parts of degree 6 or less, or use a different tool.
Cases that are not errors
The following expressions are valid inputs. If they cannot be factored further over the integers, they will simply be shown as-is in the result area.
x^2+1
x^2+xy+y^2
x^2+5y^3+3
Precautions
- Expressions with multiple variables are supported, but full factorization works best when the expression can be reorganized as a polynomial in one main variable
- Use ^ for exponents, for example x^2 or x^3
- Expressions up to degree 6 are supported
- If an expression cannot be factored further over the integers, that message will be shown
- Do not include an equals sign in the input
Frequently Asked Questions
Q1: Why does it say the expression cannot be factorized?
A: That message appears when the expression cannot be factored any further over the integers. In some cases it may still be factorable over the real or complex numbers.
Q2: Can I use variables other than x?
A: Yes. Single-letter variables such as y or z are also supported.
Q3: In what order should I try factorization methods?
A: A good order is common factors first, then Formulas 1 to 3, and finally substitution or repeated factorization.
Q4: Does it support decimal coefficients?
A: You can enter them, but the calculator mainly checks for factorization over integer coefficients.
Q5: Does it support degree 4 or higher?
A: Yes, currently up to degree 6 is supported.