What method do I use to factor simple polynomials?
Is it a binomial, trinomial, or polynomial with 4 terms??
Perfect Squares
are 1, 4, 9, 16,
25, 36, x2, x4, etc.
Binomial
(2 terms)
Trinomial
(3 terms)
Polynomial
(4 terms)
Use Factoring by grouping
Is the first term and last term a perfect
square, and there’s a subtraction sign in
between?
1. Take the first 2 terms and use
Factoring by GCF on those.
2. Take the last 2 terms and use
Factoring by GCF on those.
3. Factor the result.
Is the leading coefficient
(l.c.) = 1 and highest
exponent 2?
(The Leading Coefficient is
the number in front of x2.
Ex. Factor 3x3 -6x2 +8x - 16
No
Yes
Use Factoring the
Difference of Two
Squares
Take the square root of
the 1st term, square root
of the last term. One
factor has + the other -.
Ex. Factor x2 – 16
(x + 4)(x – 4)
Is the first term a perfect cube and the last term a perfect
cube? Perfect cubes are 1, 8, 27, 64, 125,216, x3, x6, etc.
Yes
3x3 = 3* x * x * x
-6x2 = -1 * 3 * 2 * x * x
No
3x2(x – 2)
Yes
8x =
2*2*2*x
-16 = -1 * 2 * 2 * 2 * 2
No
8(x – 2)
Use Factoring Perfect Cubes
This can be the sum or
difference.
Sum formula
a3 + b3 = (a + b)(a2 – ab + b2)
Difference Formula
a3 – b3 = (a – b)(a2 + ab + b2)
Rewrite the problem in terms of
cubes, and identify the a and b.
Plug your a and b into the
correct formula.
3
Use Factoring by GCF
(greatest common
factor)
1. Complete the prime
factorization of all
terms.
2. Circle what’s in
common to both terms.
This is the GCF, and it
goes outside of the
parenthesis.
3. The remaining terms
go inside of the
parenthesis
Use Factoring trinomials
where the leading
coefficient = 1
Ask yourself, what 2
numbers can be multiplied
together to give you the
constant term and added
together to get the
coefficient of the linear
term?
Ex. Factor x2 + 5x + 6
EX. Factor 8x – 125
(2x)3 – (5)3, so a = 2x, b = 5, and
there is a minus sign so we use
the difference formula.
(2x– 5)(4x2 + 10x + 25)
Ex. Factor 2x2 + 14x
2*x*x
2*7*x
2x(x + 7)
What 2 numbers can I
multiply to get 6(the
constant) and add to get
5(coefficient of linear
term)?
(x + 3)(x + 2)
Use Factoring trinomials where
the leading coefficient ≠ 1
3x2(x – 2) + 8(x – 2)
1. Multiply the l.c. and the
constant term.
2. What 2 numbers can you
multiply together to get the
answer to #1 and add to get
the coefficient of the linear
term?
3. Split the linear term
4. Factor by grouping
(x – 2)( 3x2 + 8)
Ex. Factor 2x2 + 11x + 12
•
2*12 = 24
• What 2 numbers can I multiply
to get 24 and add to get 11?
• 2x2 + 8x + 3x + 12
• 2x(x + 4) + 3(x + 4)
(2x + 3)(x + 4)
ALWAYS check to see if there is a GCF first !!!!!!
Practice
1.
5x2 - 125
2.
15x2b – 10ab2
3.
3m2 – y – 10
4.
9xy – 3xy2
5.
16x2 - 4
6.
-15x – 25x2
7.
x2 – 16
8.
3x2 – 15x - 42
9.
3x2 + x – 2
10. x2 – 3x – 10
11. 9x - 3
12. x2 – 8x + 15
13. x2 – xy – 12y2
14. 2x2 - 50
15. 121x2 - 169
16. 6x2 + 8x – 8
17. 3x3 + 5x2 + 12x + 20
18. 4a2 + a – 3
19. 8m3 – 1
20. 6x2 – 17x + 5
21. y2 – 16y + 64
22. t2h + 3t
23. 64c3 – 27d3
24. x2 + 6x + 5
25. x2 – 25
26. 2x2 + 15x + 7
27. p2 – 9p
28. x2 + 2x – xy – 2y
29. y2 – 8y + 12
30. 27x3 + 64
31. 81c2 – 49
32. x2 + 3x – 18
33. 18p3 – 51p2 – 135p
34.
36. u3 – 9u2 + 18u
37. 3x2 – 12y2
38. 6x2 – 3x
39. 4x6 – 4x2
x3+ 5x2 - 3x - 15
35. 6c2 + 11c + 4
40. 25r2 + 9
ALWAYS check to see if there is a GCF first !!!!!!
Practice
1.
5x2 - 125
2.
15x2b – 10ab2
3.
3m2 – y – 10
4.
9xy – 3xy2
5.
16x2 - 4
6.
-15x – 25x2
7.
x2 – 16
8.
3x2 – 15x - 42
9.
3x2 + x – 2
10. x2 – 3x – 10
11. 9x - 3
12. x2 – 8x + 15
13. x2 – xy – 12y2
14. 2x2 - 50
15. 121x2 - 169
16. 6x2 + 8x – 8
17. 3x3 + 5x2 + 12x + 20
18. 4a2 + a – 3
19. 8m3 – 1
20. 6x2 – 17x + 5
21. y2 – 16y + 64
22. t2h + 3t
23. 64c3 – 27d3
24. x2 + 6x + 5
25. x2 – 25
26. 2x2 + 15x + 7
27. p2 – 9p
28. x2 + 2x – xy – 2y
29. y2 – 8y + 12
30. 27x3 + 64
31. 81c2 – 49
32. x2 + 3x – 18
33. 18p3 – 51p2 – 135p
34.
36. u3 – 9u2 + 18u
37. 3x2 – 12y2
38. 6x2 – 3x
39. 4x6 – 4x2
x3+ 5x2 - 3x - 15
35. 6c2 + 11c + 4
40. 25r2 + 9