1. FACTORING TRINOMIALS with leading coefficient
ax2 + bx + c to
(ax + b)(cx + d)

2. REMEMBER
There are many different methods to use when factoring. To be consistent, we will continue to use the factor by grouping method.
First, use parentheses to group terms with common factors.
Be sure middle sign is +
If it is -, change it to +(-)
Next, factor the GCF from each grouping.
Now, Distributive Property…. Group both GCF’s. and bring down one of the other ( ) since they’re both the same.

3. Using grouping with trinomials
Multiply the first and last terms.
21c2(-4) =-84c2
Check your signs...
‘c’ is – so subtract factors.
‘b’ is – so larger factor will be negative.
Replace the middle term with these factors..
Then group.
Using grouping with trinomials
Switch factors to
make middle sign +
Factors of -84
(larger is - )
Difference of
factors = -5

4. Using grouping with trinomials
Multiply the first and last terms.
25y2(9) = 225y2
Check your signs...
‘c’ is + so add factors.
‘b’ is – so both factors will be negative.
Replace the middle term with these factors..
Then group.
Using grouping with trinomials
Factors of 225
(both are - )
Sum of
factors = -30
If the first term in the ( ) is negative,
You will factor out a negative number.

5. Using grouping with trinomials
Multiply the first and last terms.
9x2(2) = 18x2
Check your signs...
‘c’ is + so add factors.
‘b’ is – so both factors will be negative.
Replace the middle term with these factors..
Then group.
Using grouping with trinomials
Factors of 18
(both are - )
Sum of
factors = -9
If the first term in the ( ) is negative,
You will factor out a negative number.

6. Using grouping with trinomials
Multiply the first and last terms.
12x2(5) = 60x2
Check your signs...
‘c’ is + so add factors.
‘b’ is + so both factors will be positive.
Replace the middle term with these factors..
Then group.
Using grouping with trinomials
Sum of
factors = +19
Factors of 60
(both are + )

7. Put in descending order first!!
What's different here?
It's out of order...
Put in descending order first!!
Now follow
the same steps!
Sum of
factors = +33
Factors of 200
(both are + )

8. Put in descending order first!!
What's different here?
It's out of order...
Put in descending order first!!
Now follow
the same steps!
Difference of
factors = +3
Factors of -40
(larger is + )

9. Using grouping with trinomials
Multiply the first and last terms.
4r2(7) = 28r2
Check your signs...
‘c’ is + so add factors.
‘b’ is - so both factors will be negative.
None of the factors will ADD to give -1
Using grouping with trinomials
Sum of
factors = -1
Factors of 28
(both are - )

10. Using grouping with trinomials
Multiply the first and last terms.
3x2(-5) = -15x2
Check your signs...
‘c’ is - so subtract factors.
‘b’ is + so larger factor will be positive.
None of the factors will SUBTRACT to give +7
Using grouping with trinomials
Factors of -15
(larger is + )
Difference of
factors = +7

11. Solutions to Trinomials
Now that we have the factors of each trinomial, we can carry it to the next step and find the SOLUTIONS for each trinomial.
Remember to set your factors equal to zero then solve for the variable…. Like this….

12. The roots (solutions) are -1/3 and 4/7