Factoring Trinomials: Leading Coefficient 1
Factor trinomials of the form x² + bx + c by finding two numbers that multiply to c and add to b. Develop systematic strategies for identifying factor pairs and verify by expanding.
A lesson of Algebra 1, in Factoring Polynomials, about 30 minutes.
By the end of this lesson, you will be able to factor $x^2 + bx + c$ by finding two numbers, read the signs off $b$ and $c$ before searching, know when a trinomial does not factor, and check every answer by multiplying back.
Two Numbers, Multiplying and Adding
Multiply two binomials and watch where the numbers land.
$$(x + 2)(x + 5) = x^2 + 7x + 10$$
The $10$ is $2 \times 5$. The $7$ is $2 + 5$. That is not a coincidence, it is the expansion:
$$(x + p)(x + q) = x^2 + (p + q)x + pq$$
So factoring $x^2 + bx + c$ is a single question:
$$\textbf{which two numbers multiply to } c \textbf{ and add to } b?$$
Find them and you are done. $x^2 + 7x + 10$ needs two numbers multiplying to $10$ and adding to $7$. That is $2$ and $5$, so it factors as $(x+2)(x+5)$.
Start from the multiplying, not the adding. There are only a few factor pairs of $c$ and endlessly many pairs adding to $b$, so listing the products is a short search and listing the sums is not.
The Signs Tell You Where to Look
Before searching, read the two signs. They cut the work in half.
| Sign of $c$ | Sign of $b$ | The two numbers |
|---|---|---|
| $+$ | $+$ | both positive |
| $+$ | $-$ | both negative |
| $-$ | either | one of each |
When $c$ is positive, the numbers share a sign, because only same-signed numbers multiply to a positive. Then $b$ says which sign they share.
When $c$ is negative, they have opposite signs, and $b$ tells you which one is bigger: the sign of $b$ is the sign of the number with the larger size.
$$x^2 - 5x + 6 \quad \Rightarrow \quad \text{both negative} \quad \Rightarrow \quad (x-2)(x-3)$$
$$x^2 - x - 6 \quad \Rightarrow \quad \text{one of each, negative wins} \quad \Rightarrow \quad (x+2)(x-3)$$
Example 1: The Whole Method
Here is every step on one trinomial, worked all the way through. $$x^2 + 9x + 20$$
- What must the two numbers multiply to? 20
- What must they add to? 9
- Both signs are positive, so both numbers are positive. How many factor pairs does 20 have? 3
- One of those pairs adds to 9. Type the smaller number of that pair 4
- And the larger one? 5
- Write the factored form (x + 4)(x + 5)
Worked Example 1: Both Negative
$$x^2 - 7x + 12$$