Factoring Trinomials (a=1)
Learn to factor quadratic trinomials where the leading coefficient is 1.
Definition
- Sum: (the coefficient of )
- Product: (the constant term)
Try it now
Worked Examples
Factor:
Identify what we need
Find two numbers that ADD to 7 and MULTIPLY to 12 → Sum = 7, Product = 12
List factor pairs of 12
, , → Three possibilities
Check which pair sums to 7
(no), (no), (yes!) → ,
Write the factored form
→
Verify by expanding
\checkmark → Correct!
Answer:
Common Mistakes
Confusing sum and product requirements
Why it's wrong: Students sometimes look for numbers that multiply to and add to , when it should be the opposite.
Correct: Remember: the numbers ADD to the middle coefficient () and MULTIPLY to the constant ().
Forgetting to consider negative factors
Why it's wrong: When is positive but is negative, both factors must be negative.
Correct: Use the sign rules: positive product means same signs, negative product means different signs.
Writing as
Why it's wrong: This error comes from adding the constants instead of properly expanding.
Correct: Always verify by FOILing:
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: has no integer factors.
Interactive Visual
Interactive Sandbox
Expression Calculator
Try these:
History
No calculations yet
Practice Problems
16 problemsWhat two numbers add to 7 and multiply to 12?
Why It Matters
- 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
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 . Factoring shows it lands at seconds.
A rocket's height above ground is modeled by (in meters, after seconds).
At what times is the rocket at ground level?
Step 1: Write the mathematical expression
Factor:
Garden Design
Landscape architects use factoring to determine dimensions when given area constraints.
Example:
A garden's area is square meters. Factoring gives , revealing possible dimensions.
A rectangular pool has an area of square meters.
What are the dimensions of the pool in terms of ?
Step 1: Write the mathematical expression
Factor:
Key Takeaways
- 1For , find two numbers that ADD to and MULTIPLY to
- 2If and : both factors are positive
- 3If and : both factors are negative
- 4If : one factor is positive, one is negative (larger has the sign of )
- 5Always verify your answer by expanding with FOIL
Frequently Asked Questions
Glossary
- Trinomial
- A polynomial with exactly three terms (e.g., )
- 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 )
- Prime polynomial
- A polynomial that cannot be factored using integers
Formula Card
Factoring Pattern
where $m + n = b$ and $m \times n = c$
Sign Rules
The sign of $b$ determines which factor is larger when signs differ