Factoring Polynomials Calculator - Step-by-Step Solutions

Enter a polynomial in one or more variables. The calculator writes it as a product of factors with whole coefficients that cannot be factored any further, and shows how to get there step by step.

Factor

How to enter a polynomial

• Write powers with ^: x^2 is $x^2$ and (x+1)^3 is $(x+1)^3$.
• The multiplication sign can be left out: 3xy, 2(x+1), (x+1)(x-2). You can also write 3*x*y.
• Every letter is a variable of its own, so xy means ${x \cdot y}$.
• Coefficients can be fractions or decimals: x^2/4 - 1, 0.5x^2 - 2.
• Brackets are multiplied out first, so you can also check your own factoring: (x+2)(x+3) - 2.

Examples: $x^2 - 5x + 6$ $x^4 - 16$ $6x^2 + 7x - 3$ $x^3 - 6x^2 + 11x - 6$ $ax + ay + bx + by$

To divide polynomials, use the Polynomial Long Division Calculator.

Factoring methods

To factor a polynomial means to write it as a product of polynomials of lower degree. The calculator uses the same methods you learn at school, in this order. Click an example to see the method at work.

1. Common factor. Take out the greatest number and the highest power of each variable that divide every term.
   ${6x^2y - 9xy^2} = {3xy(2x - 3y)}$
   Example: $3x^3 - 12x$
2. Difference of squares. A sum of squares ${a^2 + b^2}$ cannot be factored.
   ${a^2 - b^2} = {(a - b)(a + b)}$
   Example: $x^4 - 16$
3. Perfect square trinomial.
   ${a^2 + 2ab + b^2} = {(a + b)^2}$
   ${a^2 - 2ab + b^2} = {(a - b)^2}$
   Example: $4x^2 - 12xy + 9y^2$
4. Sum and difference of cubes.
   ${a^3 + b^3} = {(a + b)(a^2 - ab + b^2)}$
   ${a^3 - b^3} = {(a - b)(a^2 + ab + b^2)}$
   Example: $8x^3 + 27$
5. Quadratic trinomial. For ${x^2 + bx + c}$ find two numbers $m$ and $n$ whose product is $c$ and whose sum is $b$. Then
   ${x^2 + bx + c} = {(x + m)(x + n)}$
   For ${ax^2 + bx + c}$ find two numbers whose product is ${a \cdot c}$ and whose sum is $b$, split the middle term with them and factor by grouping. If the discriminant ${D = b^2 - 4ac}$ is negative or is not a perfect square, the trinomial cannot be factored with rational coefficients.
   Examples: $x^2 - 5x + 6$, $6x^2 + 7x - 3$, $x^2 + x + 1$
6. Grouping. Split the terms into groups that have the same factor in brackets.
   ${ax + ay + bx + by} = {a(x + y) + b(x + y)} = {(a + b)(x + y)}$
   Examples: $x^3 - 2x^2 - 9x + 18$, $x^2 + 2xy + y^2 - z^2$
7. Rational roots. If a polynomial with whole coefficients has a rational root $\frac{p}{q}$, then $p$ divides the constant term and $q$ divides the leading coefficient. When ${x = r}$ is a root, ${x - r}$ is a factor, and dividing by it (Horner's method) leaves a polynomial of lower degree.
   Example: $x^3 - 6x^2 + 11x - 6$
8. Completing the square. Add and subtract a term to get a difference of squares.
   ${x^4 + 4} = {(x^4 + 4x^2 + 4) - 4x^2} = {(x^2 + 2)^2 - (2x)^2}$
   Example: $x^4 + 4$

The factors have rational coefficients. So ${x^2 - 2}$ is left as it is: it equals ${(x - \sqrt{2})(x + \sqrt{2})}$, but $\sqrt{2}$ is not a rational number.

Other resources involving factoring

Factoring - formulas Factoring polynomials - problems with solutions Simplifying polynomial expressions - problems with solutions Polynomial Long Division Calculator