Factor Any Expression

Look, we know it's 2025 and you're ready for step-by-step math help using the power of AI tools. We've been doing it, too, and AI does offer some compelling advantages with math tutoring in particular. The tool I've been playing with lately is Solvely.ai. Solvely has classic "solver" tools that can help with step-by-step guidance on tough problems, but offers additional study help with automatic quiz generation, flash cards, and study guides.

Remember that this is a tool, just like having a math tutor sitting at your side. The best way to really learn is to work through your homework problems yourself with paper and pencil, and to reach out for help when you need it. Don't rely on any one tool to do all the work for you, and always check the solution for correctness!

Try the Algebra Calculator to factorize your expressions:

Factoring Calculator

How does Solvely.ai work?

Enter your problem in the Solvely.ai website above and click SOLVE to submit your question. Once you get the hang of factoring expressions you can use the same link to solve all kinds of problems. If you're not sure what to enter, look over the sample problems below to see the types of expressions this tool can factorise.

Ok, but remind me how factorization works first:

Take a look at these five polynomials that have been factored in different ways.

Example 1: Simple Trinomial (Leading Coefficient = 1)

Problem: Factor \(x^2 + 7x + 12\)

Solution:

We need to find two numbers that multiply to give 12 and add to give 7.

Factors of 12: 1 and 12, 2 and 6, 3 and 4

Checking sums:

  • \(1 + 12 = 13\) ✗
  • \(2 + 6 = 8\) ✗
  • \(3 + 4 = 7\) ✓

Therefore: \(x^2 + 7x + 12 = (x + 3)(x + 4)\)

Check: \((x + 3)(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12\) ✓

Example 2: Trinomial with Leading Coefficient > 1

Problem: Factor \(3x^2 + 11x + 6\)

Solution:

We need two numbers that multiply to \((3)(6) = 18\) and add to \(11\).

Factors of 18: 1 and 18, 2 and 9, 3 and 6

Checking sums:

  • \(2 + 9 = 11\) ✓

Now rewrite the middle term using 2 and 9:

\(3x^2 + 11x + 6 = 3x^2 + 2x + 9x + 6\)

Factor by grouping:

  • Group 1: \(3x^2 + 2x = x(3x + 2)\)
  • Group 2: \(9x + 6 = 3(3x + 2)\)

Therefore: \(3x^2 + 11x + 6 = x(3x + 2) + 3(3x + 2) = (x + 3)(3x + 2)\)

Check: \((x + 3)(3x + 2) = 3x^2 + 2x + 9x + 6 = 3x^2 + 11x + 6\) ✓

Example 3: Difference of Squares

Problem: Factor \(4x^2 - 25\)

Solution:

Recognize this as a difference of squares: \(a^2 - b^2 = (a + b)(a - b)\)

Here, \(4x^2 = (2x)^2\) and \(25 = 5^2\)

Therefore: \(4x^2 - 25 = (2x)^2 - 5^2 = (2x + 5)(2x - 5)\)

Check: \((2x + 5)(2x - 5) = 4x^2 - 10x + 10x - 25 = 4x^2 - 25\) ✓

Example 4: Trinomial with Negative Terms

Problem: Factor \(x^2 - 5x - 24\)

Solution:

We need two numbers that multiply to \(-24\) and add to \(-5\).

Since the product is negative, one number must be positive and one negative.

Factors of 24: 1 and 24, 2 and 12, 3 and 8, 4 and 6

Checking with appropriate signs:

  • \(-8 + 3 = -5\) ✓
  • These multiply to \((-8)(3) = -24\) ✓

Therefore: \(x^2 - 5x - 24 = (x - 8)(x + 3)\)

Check: \((x - 8)(x + 3) = x^2 + 3x - 8x - 24 = x^2 - 5x - 24\) ✓

Example 5: Factoring Out GCF First

Problem: Factor \(6x^3 + 18x^2 - 24x\)

Solution:

Step 1: Find the greatest common factor (GCF).

  • All terms have a factor of 6 and at least one \(x\)
  • GCF = \(6x\)

Step 2: Factor out the GCF:

\(6x^3 + 18x^2 - 24x = 6x(x^2 + 3x - 4)\)

Step 3: Factor the remaining trinomial \(x^2 + 3x - 4\).

We need two numbers that multiply to \(-4\) and add to \(3\):

  • \(4 + (-1) = 3\) ✓
  • \(4 \cdot (-1) = -4\) ✓

Therefore: \(x^2 + 3x - 4 = (x + 4)(x - 1)\)

Final Answer: \(6x^3 + 18x^2 - 24x = 6x(x + 4)(x - 1)\)

Check: \(6x(x + 4)(x - 1) = 6x(x^2 - x + 4x - 4) = 6x(x^2 + 3x - 4) = 6x^3 + 18x^2 - 24x\) ✓

I still need more help. How would I enter problems like those into a tool like Solvely?

Let's look back at the example example we just worked. The task was to factor \(6x^3 + 18x^2 - 24x\). It's formatted nicely using something called LaTeX, but a common convention for typing math using traditional characters is to use plenty of parenthesis, the symbols * and / for multiplication and division, and the caret (^) for exponents.

For example, to factor \(6x^3 + 18x^2 - 24x\) we might ask Solvely's Algebra Calculator to "factor 6x^3 + 18x^2 - 24x"

Try typing these expressions into the Solvely tool yourself, ask it to factor them, and click SOLVE to see a demonstration. Or, use these as a template to create and solve your own problems.

Problem: \(4x^2-9\)

  • Solvely Input: "Factor 4x^2-9"
  • Expected Answer: \((2x+3)(2x-3)\)

Problem: \(x^4-81\)

  • Solvely Input: "Factor x^4-81"
  • Expected Answer: \((x^2+9)(x+3)(x-3)\)

Problem: \(x^2-7x-18\)

  • Solvely Input: "Factor x^2-7x-18"
  • Expected Answer:\((x-9)(x+2)\)

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Common Factoring Questions

Here are some questions other visitors have asked on our free math help message board. Perhaps you can learn from the questions someone else has already asked.

Final Thoughts

You may want to read up on the quadratic formula to help your algebra knowledge rather than relying on this solver. Afterall, the point is to learn the concept, not just get the answer... right?

Also, while this calculator page is tailored for algebraic expressions, you might be looking to solve for the prime factorization of a number. For example, finding all the prime numbers that divide into 56 (7 and 2). We also have a page on the greatest common factor and a link for least common multiple available.

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