Factor each of the following expressions completely and then simplify, if possible:
a: ax + 2x
b: ax - 2x
c: 3x - 4x + 7x
d: 3x^2 + xy - x
e: (a + b)(c + 1) - (a + b)
f: x^2 - 9y^2
g: (x^2 - y^2) + x + y
No ideas on any of them?
Look for groupings of like expressions, difference of two squares, etc.
For example, (g) is
(x^2-y^2) + x + y
(x-y)(x+y) + (x+y)
(x-y+1)(x+y)
I still don't understand how you got that though, factoring is my major weakness. I don't get how you get the answers to them.
To factor an expression completely, we want to find the common factors of the terms and group them together.
a: ax + 2x
Step 1: Both terms have a common factor of x. Factoring out x, we get: x(a + 2)
The expression is already factored completely.
b: ax - 2x
Step 1: Both terms have a common factor of x. Factoring out x, we get: x(a - 2)
The expression is already factored completely.
c: 3x - 4x + 7x
Step 1: Group the terms.
(3x - 4x) + 7x
Step 2: Each group has a common factor. Factoring out the common factors, we get:
-x(4 - 3) + 7x
Simplifying further:
-x + 7x
The expression becomes: 6x
d: 3x^2 + xy - x
Step 1: There are no common factors among the terms.
The expression is already factored completely.
e: (a + b)(c + 1) - (a + b)
Step 1: Distribute the first term.
a(c + 1) + b(c + 1) - (a + b)
Step 2: Group the terms.
(ac + a) + (bc + b) - (a + b)
Step 3: Each group has a common factor. Factoring out the common factors, we get:
a(c + 1) + b(c + 1) - (a + b)
Simplifying further:
(a + b)(c + 1) - (a + b)
The expression is already factored completely.
f: x^2 - 9y^2
Step 1: This expression is a difference of squares, which can be factored as:
(x - 3y)(x + 3y)
g: (x^2 - y^2) + x + y
Step 1: This expression is similar to the previous one. It's a sum of squares, which can be factored as:
(x + y)(x - y) + x + y
The expressions in parts a, b, d, e, and g are already factored completely. In part c, the expression simplifies to 6x. In part f, the expression factors as (x - 3y)(x + 3y).