Factoring Trinomials A 1 Answer Key

Factoring Trinomials a = 1 Answer Key: A Step‑by‑Step Guide with Practice Problems

Factoring trinomials where the leading coefficient equals 1 is a foundational skill in algebra that opens the door to solving quadratic equations, simplifying rational expressions, and understanding polynomial graphs. Mastering this technique not only boosts confidence in homework and exams but also builds the logical thinking needed for higher‑level mathematics. In this article we break down the process, provide clear examples, and include an answer key for a set of practice problems so you can check your work instantly.

Understanding Trinomials with a = 1

A trinomial is a polynomial with three terms, generally written in the form

[ ax^{2}+bx+c ]

When a = 1, the expression simplifies to

[ x^{2}+bx+c ]

Our goal is to rewrite this trinomial as a product of two binomials:

[ x^{2}+bx+c = (x + m)(x + n) ]

where m and n are numbers that satisfy two conditions:

1. m + n = b (the coefficient of the x‑term)
2. m · n = c (the constant term)

Finding the correct pair (m, n) is the heart of factoring when a = 1.

Step‑by‑Step Procedure

Follow these steps to factor any trinomial of the form (x^{2}+bx+c):

1. Identify b and c – Write down the coefficient of x (b) and the constant term (c).
2. List factor pairs of c – Find all integer pairs whose product equals c. Remember to consider both positive and negative pairs because c may be negative.
3. Select the pair that sums to b – From the list, choose the pair whose sum equals b.
4. Write the binomials – Insert the chosen numbers into ((x + m)(x + n)).
5. Check your work – Expand the binomials (FOIL) to verify you obtain the original trinomial.

If no integer pair satisfies both conditions, the trinomial is prime over the integers (it cannot be factored further using whole numbers).

Worked Examples

Example 1: (x^{2}+5x+6)

1. b = 5, c = 6

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2. Factor pairs of 6: (1, 6), (2, 3), (‑1, ‑6), (‑2, ‑3)

3. Pair that sums to 5: (2, 3) because 2 + 3 = 5

4. Factored form: ((x+2)(x+3)) 5. Check: ((x+2)(x+3)=x^{2}+3x+2x+6=x^{2}+5x+6) ✔️ ### Example 2: (x^{2}-4x-12)

5. b = ‑4, c = ‑12

6. Factor pairs of –12: (1, ‑12), (‑1, 12), (2, ‑6), (‑2, 6), (3, ‑4), (‑3, 4)

7. Pair that sums to –4: (2, ‑6) because 2 + (‑6) = ‑4

8. Factored form: ((x+2)(x-6))

9. Check: ((x+2)(x-6)=x^{2}-6x+2x-12=x^{2}-4x-12) ✔️

Example 3: (x^{2}+x+1)

1. b = 1, c = 1 2. Factor pairs of 1: (1, 1), (‑1, ‑1)
2. Neither pair sums to 1 (1+1=2, ‑1+‑ׁ=‑2)
3. Result: The trinomial is prime over the integers.

Practice Problems

Factor each trinomial completely. If it cannot be factored using integers, write “prime”.

Answer Key