5.2 Factoring Trinomials (a = 1)
Learning Objectives
By the end of this section, you should be able to:
- Recognize and factor trinomials of the form x² + bx + c.
- Factor trinomials with a = 1 by finding two numbers that multiply to c and add to b.
- Identify and factor out a GCF before factoring when needed.
- Factor multivariable trinomials.
- Identify prime trinomials that cannot be factored using integers.
Recognizing the Pattern
A trinomial of the form x² + bx + c factors into
(x + m)(x + n)
where:
- m × n = c (the constant term)
- m + n = b (the coefficient of x)
Tip: Signs matter! Think about which combinations of positives and negatives multiply to c and add to b.
Prime Trinomials: If no two integers satisfy m × n = c and m + n = b, the trinomial is prime.
Example 5.2.0
Factor x² + 4x + 12.
Solution
No integer factors multiply to 12 and add to 4, so this trinomial is prime.
Example 5.2.1
Factor x² + 7x + 10.
Solution
Multiply = 10, Add = 7 → 2 and 5
x² + 7x + 10 = (x + 2)(x + 5)
Example 5.2.2
Factor x² − 9x + 20.
Solution
Multiply = 20, Add = −9 → −4 and −5
(x − 4)(x − 5)
Example 5.2.3
Factor x² + 2x − 15.
Solution
Multiply = −15, Add = 2 → 5 and −3
(x + 5)(x − 3)
Example 5.2.4
Factor x² − 13x + 36.
Solution
Multiply = 36, Add = −13 → −9 and −4
(x − 9)(x − 4)
Example 5.2.5
Factor x² + x + 10.
Solution
No integers multiply to 10 and add to 1, so this trinomial is prime.
Example 5.2.6
Factor y² + 8y + 12.
Solution
Multiply = 12, Add = 8 → 2 and 6
(y + 2)(y + 6)
Example 5.2.7
Factor a² − a − 6.
Solution
Multiply = −6, Add = −1 → −3 and 2
(a − 3)(a + 2)
Try It
Factoring Out a GCF First
Before using the trinomial pattern, always check for a greatest common factor. Factoring it out first simplifies the trinomial.
Example 5.2.8
Factor 4x² + 12x − 40.
Solution
GCF = 4
4x² + 12x − 40 = 4(x² + 3x − 10)
Multiply = −10, Add = 3 → 5 and −2
4(x + 5)(x − 2)
Example 5.2.9
Factor 2t³ + 8t² + 6t.
Solution
GCF = 2t
2t³ + 8t² + 6t = 2t(t² + 4t + 3)
Multiply = 3, Add = 4 → 1 and 3
2t(t + 1)(t + 3)
Example 5.2.10
Factor −3x² − 6x + 27.
Solution
GCF = −3
−3x² − 6x + 27 = −3(x² + 2x − 9)
No integers multiply to −9 and add to 2, so x² + 2x − 9 is prime.
Final: −3(x² + 2x − 9)
Try It
Multivariable Trinomials (a = 1)
When a trinomial has two variables, treat the second variable as part of the coefficient and factor as if it were a single-variable trinomial.
Example 5.2.11
Factor a² + 9ab + 20b².
Solution
Multiply = 20b², Add = 9b → 4b and 5b
(a + 4b)(a + 5b)
Example 5.2.12
Factor x² − 3xy − 10y².
Solution
Multiply = −10y², Add = −3y → −5y and 2y
(x − 5y)(x + 2y)
Example 5.2.13
Factor a² + 5ab + 11b².
Solution
No integer factors multiply to 11b² and add to 5b, so this trinomial is prime.
Try It
Concept Check
Key Takeaways
- When a = 1, find two numbers that multiply to c and add to b.
- Always factor out the GCF first.
- The same pattern works for multivariable trinomials like a² + 9ab + 20b².
- If no integer pair works, the polynomial is prime.
- Check your factors by expanding to verify they match the original trinomial.
