Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for algebra 1 factoring polynomials? Then this Algebra 1 lesson on Factoring Polynomials with Patterns and Grouping explains each idea step by step, with worked examples.
Quick answer: Factoring rewrites a polynomial as a product of simpler polynomials. Start by taking out the greatest common factor, then look for a pattern: a difference of squares, a perfect square trinomial, a trinomial with two integers that fit, or grouping. Factoring helps solve equations because a product is zero only when one of its factors is zero.
Standards: A-SSE.A.2, A-SSE.B.3a · Level: Grade 9 Algebra 1
What does it mean to factor a polynomial?
Factoring rewrites a sum as a product. Because 6x² + 9x = 3x(2x + 3), the product form shows the pieces that multiply together, and it shows where a graph crosses the x-axis.
The largest monomial that divides every term. Take it out first, every time: 12x³ + 18x² = 6x²(2x + 3).
How do you factor x² + bx + c?
Find two integers whose product is c and whose sum is b. For x² + 7x + 12 the pair 3 and 4 works, so the factors are (x + 3)(x + 4).
How do you factor when the leading coefficient is not 1?
- Multiply a and c. For 6x² + 11x − 10 that is −60.
- Find two numbers with product −60 and sum 11: 15 and −4.
- Split: 6x² + 15x − 4x − 10.
- Group and factor each pair: 3x(2x + 5) − 2(2x + 5).
- Write the shared factor once: (2x + 5)(3x − 2).
Which patterns factor quickly?
a² − b² = (a + b)(a − b)
a² + 2ab + b² = (a + b)²
a² − 2ab + b² = (a − b)²
Four terms often factor by grouping: x³ + 4x² + 3x + 12 = x²(x + 4) + 3(x + 4) = (x² + 3)(x + 4).
A sum of squares such as x² + 16 does not factor over the real numbers, so do not write (x + 4)².
How does factoring find zeros?
If a product equals zero, one of its factors is zero (the zero-product property). To solve x² − 2x − 8 = 0, factor to (x − 4)(x + 2) = 0, so x = 4 or x = −2. These are the x-intercepts of the graph.
Always multiply your factors back out. If you do not get the original expression, one sign is wrong.
- Take out the greatest common factor before anything else.
- For x² + bx + c, find two integers with product c and sum b.
- a² − b² = (a + b)(a − b) and a² ± 2ab + b² = (a ± b)².
- Four terms often factor by grouping in pairs.
- Zero-product property: if (x − r)(x − s) = 0 then x = r or x = s.
Factoring Polynomials with Patterns and Grouping: frequently asked questions
What should you try first when factoring?
Look for a greatest common factor of all the terms and take it out. Doing this first makes the remaining polynomial smaller and easier to factor.
How do you factor x² + bx + c?
Find two integers that multiply to c and add to b. Then write (x + p)(x + q) using those two integers. Multiply back to check the middle term.
Why can you not factor x² + 25?
A sum of two squares has no pair of real numbers that multiply to 25 and add to 0. Only a difference of squares, such as x² − 25, factors.
What is the zero-product property?
If two numbers multiply to give zero, at least one of them is zero. For (x − 3)(x + 7) = 0, set each factor equal to zero to get x = 3 or x = −7.
How can you check a factoring answer?
Multiply the factors using FOIL or an area model. The product must match the original polynomial exactly, including every sign.
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