Lesson 3: Factor Trinomials of the Form ax2+bx+c

New Concept

This lesson teaches you how to factor more complex trinomials of the form $ax^2+bx+c$. You'll master two powerful techniques—trial and error and the 'ac' method—while always remembering to first check for a Greatest Common Factor (GCF).

What’s next

Next, you'll apply these strategies through interactive examples and practice cards to build your factoring skills.

Property

Choose a strategy to factor polynomials completely.

Step 1. Is there a greatest common factor? Factor it out.

Step 2. Is the polynomial a binomial, trinomial, or are there more than three terms?

• If it is a binomial, right now we have no method to factor it.
• If it is a trinomial of the form $x^2 + bx + c$: Undo FOIL $(x \quad)(x \quad)$.
• If it has more than three terms: Use the grouping method.

Property

When factoring a polynomial, always ask first, “Is there a greatest common factor?” If there is, factor it first. This is the initial step in the comprehensive factoring strategy. Factoring out the GCF simplifies the remaining polynomial, often revealing a more familiar structure like a simple trinomial, which can then be factored further.

Examples

• Factor completely $3x^2 + 15x + 18$. The GCF is 3. Factoring it out gives $3(x^2 + 5x + 6)$. The trinomial factors to $(x+2)(x+3)$. The final answer is $3(x+2)(x+3)$.

• Factor completely $5y^2 - 15y - 50$. The GCF is 5. This gives $5(y^2 - 3y - 10)$. The trinomial factors to $(y-5)(y+2)$. The final answer is $5(y-5)(y+2)$.

Property

How to Factor Trinomials of the Form $ax^2 + bx + c$ Using Trial and Error

Step 1. Write the trinomial in descending order of powers.
Step 2. Find all the factor pairs of the first term, $ax^2$.
Step 3. Find all the factor pairs of the third term, $c$.
Step 4. Test all possible combinations of the factors. The correct combination is the one where the product of the Outer and Inner terms of the binomials sums to the middle term, $bx$.
Step 5. Check your answer by multiplying the factors.

Examples

• Factor $2x^2 + 7x + 3$. The factors of $2x^2$ are $x, 2x$. The factors of 3 are 1, 3. Testing combinations, $(x+3)(2x+1)$ gives $x+6x=7x$. So the answer is $(x+3)(2x+1)$.

Property

How to Factor Trinomials of the Form $ax^2 + bx + c$ Using the “ac” Method

Step 1. Factor out any GCF.
Step 2. Find the product $ac$.
Step 3. Find two numbers, $m$ and $n$, that multiply to $ac$ and add to $b$.
Step 4. Split the middle term into two terms using $m$ and $n$: $ax^2 + mx + nx + c$.
Step 5. Factor the resulting four-term polynomial by grouping.
Step 6. Check by multiplying the factors.

Examples

• Factor $4x^2 + 11x + 6$. Here $ac = 4 \cdot 6 = 24$ and $b=11$. Two numbers that multiply to 24 and add to 11 are 3 and 8. Rewrite as $4x^2 + 3x + 8x + 6$. Grouping gives $x(4x+3) + 2(4x+3) = (x+2)(4x+3)$.