How to Factor Trinomials When a≠1a\neq 1a=1
Factoring trinomials of the form ax2+bx+cax^2+bx+cax2+bx+c where a≠1a\neq 1a=1 can seem tricky at first, but with a systematic approach, it becomes manageable. Here's a step-by-step guide:
Step 1: Understand the trinomial form
You have a quadratic trinomial:
ax2+bx+cax^2+bx+cax2+bx+c
where aaa, bbb, and ccc are constants, and a≠1a\neq 1a=1.
Step 2: Multiply aaa and ccc
Calculate the product:
ac=a×cac=a\times cac=a×c
Step 3: Find two numbers that multiply to acacac and add to bbb
Look for two integers mmm and nnn such that:
m×n=acm\times n=acm×n=ac
and
m+n=bm+n=bm+n=b
Step 4: Rewrite the middle term using mmm and nnn
Express the middle term bxbxbx as:
mx+nxmx+nxmx+nx
So the trinomial becomes:
ax2+mx+nx+cax^2+mx+nx+cax2+mx+nx+c
Step 5: Factor by grouping
Group the terms in pairs:
(ax2+mx)+(nx+c)(ax^2+mx)+(nx+c)(ax2+mx)+(nx+c)
Factor out the greatest common factor (GCF) from each group:
x(ax+m′)+k(ax+m′)x(ax+m')+k(ax+m')x(ax+m′)+k(ax+m′)
where m′m'm′ and kkk are the factors you extract.
Step 6: Factor out the common binomial
Now, factor out the common binomial factor:
(ax+m′)(x+k)(ax+m')(x+k)(ax+m′)(x+k)
Example
Factor 6x2+11x+36x^2+11x+36x2+11x+3:
- a=6a=6a=6, b=11b=11b=11, c=3c=3c=3
- ac=6×3=18ac=6\times 3=18ac=6×3=18
- Find two numbers that multiply to 18 and add to 11: 9 and 2
- Rewrite:
6x2+9x+2x+36x^2+9x+2x+36x2+9x+2x+3
- Group:
(6x2+9x)+(2x+3)(6x^2+9x)+(2x+3)(6x2+9x)+(2x+3)
- Factor each group:
3x(2x+3)+1(2x+3)3x(2x+3)+1(2x+3)3x(2x+3)+1(2x+3)
- Factor out the common binomial:
(3x+1)(2x+3)(3x+1)(2x+3)(3x+1)(2x+3)
Tips:
- If you can't find integers mmm and nnn, the trinomial may be prime (not factorable over integers).
- Always check your factorization by expanding the factors to ensure you get the original trinomial.
If you want, I can provide more examples or help with specific problems!
