Lesson 6: Factoring ax² + bx + c

Property

• Trinomials of the form $ax^2 + bx + c$ where $a \neq 1$: Use the trial and error method or the 'ac' method to find factors.
• Trial and Error Method: Test different factor combinations of the form $(px + m)(qx + n)$ where $pq = a$ and $mn = c$.
• AC Method: Find two numbers that multiply to $ac$ and add to $b$, then rewrite the middle term and factor by grouping.

Examples

Property

When factoring expressions, first identify and extract any common binomial factors before applying other factoring methods.
A binomial common factor appears in every term of the expression: $ab + ac = a(b + c)$ where $a$ can be a binomial.

Examples

Property

How to factor trinomials of the form $ax^2 + bx + c$ using trial and error.

1. Write the trinomial in descending order of degrees.
2. Factor any GCF. If the leading coefficient is negative, the GCF will be negative.
3. Find all the factor pairs of the first term ($ax^2$). These will be the first terms of your binomial factors.
4. Find all the factor pairs of the third term ($c$). These will be the last terms.
5. Test all possible combinations of the factors. The correct combination is the one where the sum of the inner and outer products equals the middle term ($bx$).
6. Check by multiplying.

Examples

• To factor $2x^2 + 9x + 10$, the factors of $2x^2$ are $(x, 2x)$. The factors of 10 are $(1, 10)$ and $(2, 5)$. Testing combinations, we find $(x+2)(2x+5)$ gives $5x + 4x = 9x$. So, the factors are $(x+2)(2x+5)$.
• To factor $6y^2 - 19y + 10$, we need two negative factors for 10. Factors of $6y^2$ include $(2y, 3y)$. Testing combinations, $(2y-5)(3y-2)$ gives $-4y - 15y = -19y$. So, the factors are $(2y-5)(3y-2)$.
• To factor $12x^3 + 4x^2 - 16x$, first factor out the GCF, $4x$. This gives $4x(3x^2 + x - 4)$. Factoring the trinomial, we get $4x(3x+4)(x-1).

Explanation

This method is a systematic puzzle. You list all possible factors for the first ($a$) and last ($c$) coefficients and test pairs. Keep trying combinations until the 'Outer' and 'Inner' products from FOIL add up to the middle term.

Property

To factor a quadratic trinomial of the form $ax^2 + bx + c$, we reverse the FOIL method. We look for two binomials whose product matches the trinomial.

• The First terms of the binomials must multiply to $ax^2$.
• The Last terms must multiply to $c$.
• The sum of the Outer and Inner products must equal the middle term, $bx$.

Examples

• To factor $2x^2 - 7x + 3$, the first terms could be $2x$ and $x$. The last terms could be $-1$ and $-3$. The combination $(2x - 1)(x - 3)$ gives a middle term of $-6x - x = -7x$, so it is correct.