Algebra - FACTORING A NUMBER

TO FACTOR A NUMBER or an expression, means to write it as a product of factors.

Example 1 Factor 30

SOLUTION: 30 = 2• 15 = 2• 3• 5

If we begin 30 = 5• 6, we still obtain -- apart from the order-- 30 = 5• 2• 3.

SOLUTION: 50 = 2• 25 = 2• 5• 5

Factoring, then, is the reverse of multiplying. When we multiply, we write
2(a + b) = 2a + 2b.

But if we switch sides and write 2a + 2b = 2(a + b),

then we have factored 2a + 2b as the product 2(a + b).

In the sum 2a + 2b, 2 is a common factor of each term. It is a factor of 2a, and it is a factor of 2b. This Lesson is concerned exclusively with recognizing common factors.

Problem 2 Factor 3x − 3y.

SOLUTION: 3x − 3y = 3(x − y)

Problem 3 Rewrite each of the following as the product of 2x and another factor. The first one is done for you.

a) 8x = SOLUTION: 2x• 4

b) 6ax = SOLUTION: ___________

c) 2x² = SOLUTION: ___________

d) 2x to the 3rd power = SOLUTION: ___________

e) 4x to the 10th power = SOLUTION: ___________

f) 6x to the 5th power = SOLUTION: ____________

g) 2ax to the 6th power = SOLUTION: _____________

h) 2x = SOLUTION: _______________

SOLUTION: 5 is a common factor of each term. Display it on the left of the parentheses:

10a − 15b + 5 = 5(2a − 3b + 1)

If we multiply the right-hand side, we will get the left-hand side. In that way, the student can always check factoring.

Also, the sum on the left has three terms. Therefore, the sum in parentheses must also have three terms -- and it should have no common factors.

Problem 4 Factor each sum. Pick out the common factor. Check your answer. The first one is done for you.

a) 4x + 6y = SOLUTION: 2(2x + 3y)

b) 6x − 6 = SOLUTION: _______________

c) 8x + 12y − 16z = SOLUTION: __________________

d) 12x + 3 = SOLUTION: ___________________

e) 18x − 30 = SOLUTION: ____________________

f) 2x + ax = SOLUTION: ______________________

g) x² + 4x = SOLUTION: ___________________

h) 8x² − 4x = SOLUTION: _____________________