Factoring Trinomials (a=1)

Learn to factor quadratic trinomials where the leading coefficient is 1.

Intermediate25 minLesson
Printable worksheet

Definition

A trinomial is a polynomial with three terms. When factoring trinomials of the form x2+bx+cx^2 + bx + c, we find two binomials that multiply to give the original expression.
x2+bx+c=(x+m)(x+n)x^2 + bx + c = (x + m)(x + n)
where mm and nn are numbers that satisfy:
  • Sum: m+n=bm + n = b (the coefficient of xx)
  • Product: m×n=cm \times n = c (the constant term)
This method works because when we expand (x+m)(x+n)(x + m)(x + n) using FOIL:
x2+nx+mx+mn=x2+(m+n)x+mnx^2 + nx + mx + mn = x^2 + (m+n)x + mn

Try it now

What two numbers add to 7 and multiply to 12?

Worked Examples

Factor: x2+7x+12x^2 + 7x + 12
1

Identify what we need

Find two numbers that ADD to 7 and MULTIPLY to 12 → Sum = 7, Product = 12
2

List factor pairs of 12

1×121 \times 12, 2×62 \times 6, 3×43 \times 4 → Three possibilities
3

Check which pair sums to 7

1+12=131 + 12 = 13 (no), 2+6=82 + 6 = 8 (no), 3+4=73 + 4 = 7 (yes!) → m=3m = 3, n=4n = 4
4

Write the factored form

(x+3)(x+4)(x + 3)(x + 4)
5

Verify by expanding

x2+4x+3x+12=x2+7x+12x^2 + 4x + 3x + 12 = x^2 + 7x + 12 \checkmark → Correct!

Common Mistakes

Confusing sum and product requirements

Why it's wrong: Students sometimes look for numbers that multiply to bb and add to cc, when it should be the opposite.

Correct: Remember: the numbers ADD to the middle coefficient (bb) and MULTIPLY to the constant (cc).

Forgetting to consider negative factors

Why it's wrong: When cc is positive but bb is negative, both factors must be negative.

Correct: Use the sign rules: positive product means same signs, negative product means different signs.

Writing (x+3)(x+4)(x + 3)(x + 4) as x2+34x^2 + 34

Why it's wrong: This error comes from adding the constants instead of properly expanding.

Correct: Always verify by FOILing: (x+3)(x+4)=x2+4x+3x+12=x2+7x+12(x+3)(x+4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12

Assuming all trinomials can be factored with integers

Why it's wrong: Some trinomials are prime (cannot be factored with integers).

Correct: If no integer pair works, the trinomial is prime. Example: x2+5x+3x^2 + 5x + 3 has no integer factors.

Interactive Visual

Area model: factoring a trinomial

Arrange x², the x-strips and the unit squares into one rectangle: its sides are the factors.

x2x^2
3x3x
4x4x
1212
xx
33
xx
44

Length × width = area

x2+7x+12=(x+3)(x+4)x^2 + 7x + 12 = (x + 3)(x + 4)

Interactive Sandbox

Expression Calculator

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Practice Problems

16 problems
Problem 1 of 16
Easy
What two numbers add to 7 and multiply to 12?

Why It Matters

Factoring trinomials is essential for:
  • Solving quadratic equations: Setting each factor equal to zero gives the solutions
  • Graphing parabolas: Factored form reveals the x-intercepts
  • Simplifying expressions: Factored forms are easier to work with in fractions
  • Physics and engineering: Projectile motion, optimization problems
Mastering this skill unlocks the door to advanced algebra and calculus!

Real World Applications

Projectile Motion

When an object is thrown upward, its height follows a quadratic pattern. Factoring helps find when it hits the ground.

Example:

A ball's height is given by h=−t2+5t+6h = -t^2 + 5t + 6. Factoring −1(t2−5t−6)=−(t−6)(t+1)-1(t^2 - 5t - 6) = -(t-6)(t+1) shows it lands at t=6t = 6 seconds.

1Try It Yourself

A rocket's height above ground is modeled by h=−t2+8t−12h = -t^2 + 8t - 12 (in meters, after tt seconds).

At what times is the rocket at ground level?

Step 1: Write the mathematical expression

Factor: t2−8t+12t^2 - 8t + 12

Garden Design

Landscape architects use factoring to determine dimensions when given area constraints.

Example:

A garden's area is x2+11x+24x^2 + 11x + 24 square meters. Factoring gives (x+3)(x+8)(x + 3)(x + 8), revealing possible dimensions.

2Try It Yourself

A rectangular pool has an area of x2+10x+21x^2 + 10x + 21 square meters.

What are the dimensions of the pool in terms of xx?

Step 1: Write the mathematical expression

Factor: x2+10x+21x^2 + 10x + 21

Key Takeaways

  • 1For x2+bx+cx^2 + bx + c, find two numbers that ADD to bb and MULTIPLY to cc
  • 2If c>0c > 0 and b>0b > 0: both factors are positive
  • 3If c>0c > 0 and b<0b < 0: both factors are negative
  • 4If c<0c < 0: one factor is positive, one is negative (larger has the sign of bb)
  • 5Always verify your answer by expanding with FOIL

Frequently Asked Questions

The trinomial may be prime (cannot be factored with integers). For example, x2+5x+3x^2 + 5x + 3 has no integer factor pairs that add to 5 and multiply to 3.
The trinomial may be prime (cannot be factored with integers). For example, x2+5x+3x^2 + 5x + 3 has no integer factor pairs that add to 5 and multiply to 3.
No! (x+3)(x+4)(x + 3)(x + 4) is the same as (x+4)(x+3)(x + 4)(x + 3) due to the commutative property of multiplication.
That requires a different method (AC method or trial and error). This lesson focuses only on trinomials where a=1a = 1.

Glossary

Trinomial
A polynomial with exactly three terms (e.g., x2+5x+6x^2 + 5x + 6)
Factor
To write an expression as a product of simpler expressions
Leading coefficient
The coefficient of the highest-degree term (the number in front of x2x^2)
Prime polynomial
A polynomial that cannot be factored using integers

Formula Card

Factoring Pattern

x2+bx+c=(x+m)(x+n)x^2 + bx + c = (x + m)(x + n)

where $m + n = b$ and $m \times n = c$

Sign Rules

c>0⇒same signsc<0⇒different signsc > 0 \Rightarrow \text{same signs} \quad c < 0 \Rightarrow \text{different signs}

The sign of $b$ determines which factor is larger when signs differ

Today's challengeOne problem, three tries

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Factoring Out the GCF

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Factoring by Grouping

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Factoring Trinomials (General)

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