To factor a trinomial, typically of the form ax2+bx+cax^2+bx+cax2+bx+c, follow these key steps:
Factoring Trinomials When a=1a=1a=1 (i.e., x2+bx+cx^2+bx+cx2+bx+c)
- Identify bbb and ccc from the trinomial.
- Find two numbers that add up to bbb and multiply to ccc.
- Write the factors as (x+m)(x+n)(x+m)(x+n)(x+m)(x+n), where mmm and nnn are the two numbers found.
- Check by expanding to ensure the factors multiply back to the original trinomial.
Example: Factor x2+6x+8x^2+6x+8x2+6x+8.
- b=6b=6b=6, c=8c=8c=8
- Numbers that add to 6 and multiply to 8 are 4 and 2.
- Factors: (x+4)(x+2)(x+4)(x+2)(x+4)(x+2)
Factoring Trinomials When a≠1a\neq 1a=1 (i.e., ax2+bx+cax^2+bx+cax2+bx+c)
- Multiply aaa and ccc (product = acacac).
- Find two numbers that multiply to acacac and add to bbb.
- Rewrite the middle term bxbxbx as the sum of two terms using these numbers.
- Factor by grouping : group terms in pairs and factor out the common factor from each group.
- Factor out the common binomial factor to complete the factorization.
Example: Factor 6x2+11x+46x^2+11x+46x2+11x+4.
- a=6a=6a=6, b=11b=11b=11, c=4c=4c=4
- ac=24ac=24ac=24
- Numbers that multiply to 24 and add to 11 are 8 and 3.
- Rewrite: 6x2+8x+3x+46x^2+8x+3x+46x2+8x+3x+4
- Group: (6x2+8x)+(3x+4)(6x^2+8x)+(3x+4)(6x2+8x)+(3x+4)
- Factor each: 2x(3x+4)+1(3x+4)2x(3x+4)+1(3x+4)2x(3x+4)+1(3x+4)
- Factor common binomial: (2x+1)(3x+4)(2x+1)(3x+4)(2x+1)(3x+4)
Additional Tips
- Always check for a greatest common factor (GCF) first and factor it out before proceeding.
- For special cases, use formulas like perfect square trinomials or difference of squares when applicable.
- The "ac method" or "grouping method" is a reliable structured approach for all trinomials.
- Verify your factors by multiplying them back to the original trinomial.
This process reverses the expansion of binomials and helps simplify expressions or solve quadratic equations efficiently
