1. Section 5.4 Factoring FACTORING Greatest Common Factor,
Factor by Grouping,
Factoring Trinomials,
Difference of Squares,
Perfect Square Trinomial,
Sum & Difference of Cubes

2. Factoring—define factored form
Factor means to write a quantity as a multiplication problem
a product of the factors.
Factored forms of 18 are:

3. Factoring: The Greatest Common Factor
To find the greatest common factor of a list of numbers:
Write each number in prime factored form
Choose the least amount of each prime that occurs in each number
Multiply them together
Find the GCF of 24 & 36

4. Factoring: The Greatest Common Factor
To find the greatest common factor of a list of variable terms:
Choose the variables common to each term.
Choose the smallest exponent of each common variable.
Multiply the variables.
Find the GCF of:

5. Factoring: The Greatest Common Factor
To factor out the greatest common factor of a polynomial:
Choose the greatest common factor for the coefficients.
Choose the greatest common factor for the variable parts.
Multiply the factors.

6. Factoring: The Greatest Common Factor
each of the
following
by factoring
out the
greatest
common
factor:
5x + 5 =
4ab + 10a2 =
8p4q3 + 6p3q2 =
2y + 4y2 + 16y3 =
3x(y + 2) -1(y + 2) =

7. Factoring: The Greatest Common Factor
The answers are :

8. Factoring: by Grouping
Often used when factoring four terms.
Organize the terms in two groups of two terms.
Factor out the greatest common factor from each group of two terms.
Factor out the common binomial factor from the two groups.
Rearranging the terms may be necessary.

9. Factoring: by Grouping
Factor by grouping:
2 groups of 2 terms
Factor out the GCF
from each group of 2 terms
Factor out the
common binomial factor

10. Factoring: by Grouping

11. Factoring Trinomials—with a coefficient of 1 for the squared term
List the factors of 20:
Select the pairs from which 12
may be obtained
Write the two
binomial factors:
Check using FOIL:

12. Factoring Trinomials TIP
 If the last term of the trinomial is positive and the middle sign is positive, both binomials will have the same “middle” sign as the second term.

13. Factoring Trinomials TIP
 If the last term of the trinomial is positive and the middle sign is negative, both binomials will have the same “middle” sign as the second term.

14. Factoring Trinomials—with a coefficient of 1 for the squared term
List the factors of 22
Select the pair from
which –9 may be obtained
Write the two
binomial factors:
Check using FOIL:

15. Factoring Trinomials TIP
 If the last term of the trinomial is negative, both binomials will have one plus and one minus “middle” sign.

16. Factoring Trinomials—primes
A PRIME POLYNOMIAL cannot be factored using only integer factors.
Factor :
The factors of 5: 1 and 5.
Since –2 cannot be obtained from 1 and 5, the polynomial is prime.

17. Factoring Trinomials—2 variables
The factors of 8 are: 1,8 & 2,4, & -1,-8 & -2, -4
Choose the pairs from which
–6 can be obtained: 2 & 4
Use y in the first
position and z in the
second position
Write the two binomial
factors and
check your answer

18. Factoring Trinomials—with a GCF
If there is a greatest common factor?
If yes, factor it out first.

19. Factoring Trinomials—always check your factored form
Always check your answer with multiplication of the factors.
The check:

20. Factoring Trinomials— when the coefficient is not 1 on the squared term

21. Factoring Trinomials---use grouping

22. Factoring Trinomials---use grouping

23. Factoring Trinomials---use FOIL and Trial and Error

24. Factoring Trinomials---use FOIL and Trial and Error

25. Factoring Trinomials---use FOIL and Trial and Error

26. Factoring Trinomials---use FOIL and Trial and Error

27. Factoring Trinomials---use FOIL and Trial and Error

28. Factoring Trinomials---use FOIL and Trial and Error

29. Factoring Trinomials---with a negative GCF
Is the squared term negative?
If yes, factor our a negative GCF.

30. Special Factoring—difference of 2 squares
The following must be true:
There must be only two terms in the polynomial.
Both terms must be perfect squares.
There must be a “minus” sign between the two terms.

31. Special Factoring—difference of 2 squares
The following pattern holds true for the difference of 2 squares:

32. Special Factoring—difference of 2 squares
The pattern:

33. Special Factoring—difference of 2 squares
The pattern:

34. Special Factoring—difference of 2 squares
The pattern:

35. Special Factoring—difference of 2 squares
The pattern:

36. Special Factoring—perfect square trinomial
A perfect square trinomial is a trinomial that is the square of a binomial.

37. Special Factoring—perfect square trinomial
The first and third terms are perfect squares.
AND the middle term is twice the product of the square roots of the first and third terms
TEST THE MIDDLE TERM:

38. Special Factoring—perfect square trinomial
The patterns for a perfect square trinomial are:

39. Special Factoring—perfect square trinomial
Factor the following using the perfect square trinomial pattern:

40. Special Factoring—perfect square trinomial
Factor the following using the perfect square trinomial pattern:

41. Special Factoring—difference of two cubes
Factor using the pattern.

42. Special Factoring—sum of two cubes
Factor using the pattern.

43. Solving quadratic equation with factoring
A quadratic equation has a “squared” term.

44. Apply the Zero-Factor Property.
To Factor a Quadratic,
Apply the Zero-Factor Property.
If a and b are real numbers and if ab = 0, then a = 0 or b = 0.

45. Solving quadratic equations with factoring—Zero-Factor Property
Solve the equation:
(x + 2)(x - 8) = 0.
Apply the zero-factor property:
(x + 2) = or (x – 8) = 0
x = or x = 8

46. Solving quadratic equations with factoring—Zero-Factor Property
There are two answers for x:
-2 and 8.
Check by substituting the values calculated for x into the original equation.
(x + 2)(x - 8) = 0.
(-2 + 2)(-2 – 8) = 0 (8 + 2)(8 – 8) = 0
0 = = 0

47. Solving quadratic equations with factoring—Standard Form
To solve a quadratic equation,
Write the equation in standard form.
(Solve the equation for 0.)

48. Solving quadratic equations with factoring
To solve a quadratic equation,
Factor the quadratic expression.

49. Solving quadratic equations with factoring
To solve a quadratic equation,
Apply the Zero-Factor Property

50. Solving quadratic equations with factoring
To solve a quadratic equation,
Check your answers