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
A: x(a+2)
a: ax+2x = x(a+2)
b: ax-2x = x(a-2)
c: 3x-4x+7x = x(3-4+7) = 6x
d: 3x^2+xy-x = x(3x+y-1)
e: (a+b)(c+1)-(a+b) = ac+bc = c(a+b)
f: x^2-9y^2 = x=3y
g:(x^2 - y^2) + x + y = y-x
How did you do this??
Niezel is correct from a - e
explanation of e)
(a+b)(c+1)-(a+b) = ac+bc = c(a+b)
by common factor of (a_+b)
= (a+b)(c+1 - 1)
= c(a+b)
f)
x^2 - 9y^2 ----- standard difference of squares
= (x+3y)(x-3y)
g)
(x^2 - y^2) + x + y
= (x-y)(x+y) + (x+y) , difference of squares at the front, no damage done by putting in brackets at the end
= (x+y)(x-y + 1) --- just used a common factor of (x+y)
a: To factor the expression ax + 2x, we can first look for the greatest common factor (GCF) of the terms. In this case, the GCF is x. Factoring out x, we have x(a + 2).
b: Similarly, for the expression ax - 2x, the GCF is x. Factoring out x, we have x(a - 2).
c: In the expression 3x - 4x + 7x, we can combine like terms. The first two terms have the same variable (x), so we can subtract their coefficients: 3x - 4x = -x. Then, adding the third term, we have -x + 7x = 6x. Therefore, the expression can be simplified to 6x.
d: To factor the expression 3x^2 + xy - x, we first look for the GCF, which is 1. Factoring out the GCF, we have 1(3x^2 + xy - x). Since there are no common factors among the terms inside the parentheses, the expression is already factored.
e: To factor the expression (a + b)(c + 1) - (a + b), we can treat the term (a + b) as a common factor. Factoring it out, we have (a + b)(c + 1 - 1), which simplifies to (a + b)(c). Therefore, the completely factored expression is (a + b)(c).
f: To factor the expression x^2 - 9y^2, we recognize it as a difference of squares. The expression can be written as (x)^2 - (3y)^2, which follows the pattern a^2 - b^2 = (a + b)(a - b). Applying this pattern, we have (x + 3y)(x - 3y).
g: In the expression (x^2 - y^2) + x + y, we recognize the first two terms as a difference of squares. We can write it as (x + y)(x - y) + x + y. Then, we can combine like terms and rewrite it as (x + x + y)(x - y) + y. Simplifying further, we have (2x + y)(x - y) + y as the completely factored expression.