Factor each of the following expressions
X^2+2x+y^2
5x^2-16x+3
(X+y)^2-64
Ax+bx-ay-by
To factor expressions, we need to look for common factors or apply different factoring techniques depending on the structure of the expression.
1. Factor X^2+2x+y^2:
This expression cannot be factored further because it does not have any common factors or follow any specific factoring patterns.
2. Factor 5x^2-16x+3:
For this expression, we can use the factoring technique called "factoring by grouping".
First, multiply the coefficient of the quadratic term (5) and the constant term (3): 5 * 3 = 15.
Next, we need to find two numbers that multiply to give 15 and add up to the coefficient of the linear term (-16). In this case, the numbers are -1 and -15.
Rewrite the middle term -16x as -x - 15x:
5x^2 - x - 15x + 3
Now, we can factor by grouping:
(5x^2 - x) - (15x - 3)
Take out the common factors from each binomial:
x(5x - 1) - 3(5x - 1)
Combine like terms:
(x - 3)(5x - 1)
3. Factor (X+y)^2-64:
To factor this expression, we can use the factoring technique called "difference of squares".
The expression (X+y)^2 is in the form a^2 - b^2, where a = (X+y) and b = 8.
We can rewrite the expression as:
((X+y) + 8)((X+y) - 8)
Simplifying further gives us:
(X + y + 8)(X + y - 8)
4. Factor Ax + bx - ay - by:
This expression can be factored by grouping terms with common factors.
Taking out the common factors:
(Ax - ay) + (bx - by)
Factor out the variable from each binomial:
Ax(b - 1) + by(1 - a)
In this case, there are no further common factors between the two terms, so the expression is already factored.