1. P.4 FACTORING (التحليل) Objectives: Greatest Common Factor
Factoring Trinomial
Special Factoring
Factoring by Grouping
General Factoring

2. Like prime factorization of a number
Only integer coefficients and constants allowed

3. Methods of Factoring: Ex: Factor out the GCF.
Def: Factoring means to write a polynomial as a product of polynomials of lower degree.
Methods of Factoring:
1. Factor Out the Greatest Common Factor( GCF)
GCF = product of all prime factors raised to the smallest powers.
Ex: Factor out the GCF.
write each term as product of its prime factors
factor out the GCF = 6t2

4. 2. Try to Factor A binomial by One of the Following Special Factoring Formulas:
a. Difference of Two Squares:
Ex.
No similar rule for a sum of squares
Cannot be factored with real coefficients “prime”

5. Perfect square – perfect square

6. b. Sum/Difference of Two Cubes:
Ex.
GCF

7. a. As a perfect –square trinomial b. Using the trial method
Factorization Theorem:
The trinomial with integer coefficients a, b and c can be factored as the product of two binomial with integer coefficients if and only if
is a perfect square.

8. Ex: 25a2 – 90ac + 81c2
Check
25a2 is a perfect square. 25a2 = 5a  5a
81c2 is a perfect square. 81c2 = (-9c)  (-9c)
2(5a)(-9c) = -90ac
This is a perfect square trinomial.
25a2 – 90ac + 81c2 = (5a – 9c)2

9. Ex: Check Try Trail and Error Method
Smile! The leading coefficient is 1! Easy!
Set up for FOIL
( )( )
We know First term has to be x because only x*x = x2
We know last term has to be factors of 12 … 12,1; or 4,3; or 6,2.
(x 12)(x 1) or (x 4)(x 3) or (x 6)(x 2)

10. X2 + 8x + 12
The “+” sign before the “12” lets us know we will be adding the two factors.
The sum of the 2 factors must = +8!
Of the 3 pairs of factors only 6 and 2 have a sum of 8
The “+” sign before the 12 also lets us know both signs in the solution will be the same.

11. X2 + 8x + 12 Possibilities for solution: (x - 6)(x - 2) or…
(-6)(-2) = +12 and (6)(2) = +12
But = -8; = +8
Sooooo….
(x + 6)(x + 2) … check by FOIL

12. X2 - 7x + 12 Here again our leading coefficient is 1… (x )(x )
The last terms must be factors of 12 … 6,2; or 12,1; or 3,4.
The “+” before the 12 tells us we will be adding the 2 factors, and that the signs will be the same!
The sum of the factors must be -7!

13. X2 - 7x + 12 Of the 3 pairs of factors of 12, only 4 & 3 sum to 7
Signs must be the same, so…
(4)(3) = +12; = 7
(-4)(-3) = +12; = -7 … these are the factors we are looking for!
(x - 4)(x - 3)
Check by FOIL

14. X2 - 4x - 12
Leading coefficient is 1; we need two factors of 12 … whose difference is 4
The “-” sign in front of the 12 also tells us that the signs will be different in our solution.
Factors of 12 whose difference is 4 … 6 & 2
The “-” before the 4 lets us know the sign of the larger number (6) must be negative …
(x - 6)(x + 2)

15. 2x2 + 9x + 4
Perhaps by now we can recognize that both signs in the factors will be “+”
In this case we only have 3 possibilities
(2x + 4)(x + 1)
(2x + 1)(x + 4)
(2x + 2)(x + 2)
Check by FOIL to see which is a solution.

16. 3. Factoring by Grouping Factor mx Factor 2
Group the first two terms and the last two terms
watch
Factor mx
Factor 2

17. Ex: Factor

18. You’re
shining!