factor:
1. 6x^2y+4xy^3+10x^4y^4
2. 27m^3-8
3. 6n^8-11n^4-10
4. nx^2+x^2-9n-9
To factor these expressions, we need to look for common terms that can be factored out. Let's go through each expression step by step:
1. 6x^2y + 4xy^3 + 10x^4y^4
First, we can observe that all three terms have a common factor of 2xy. Factoring out 2xy, we get:
2xy(3x + 2y^2 + 5x^3y^3)
2. 27m^3 - 8
This expression is a difference of cubes because 27m^3 and 8 can both be expressed as cubes. The formula for factoring a difference of cubes is:
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
In this case, a = 3m and b = 2. Factoring using the formula, we have:
27m^3 - 8 = (3m - 2)(9m^2 + 6m + 4)
3. 6n^8 - 11n^4 - 10
This expression does not have any common factors. To factor it, we can look for a common binomial factor using trial and error. We can observe that 2 is a possible factor. Dividing the expression by 2, we get:
(2n - 5)(3n^8 + n^4 + 2)
So, the factored form of 6n^8 - 11n^4 - 10 is (2n - 5)(3n^8 + n^4 + 2).
4. nx^2 + x^2 - 9n - 9
This expression can be grouped into (nx^2 + x^2) - (9n + 9). Then, we can factor out the common terms within each group:
x^2(n + 1) - 9(n + 1)
Now, we have a common binomial factor of (n + 1). Factoring it out, we get:
(n + 1)(x^2 - 9)
The expression x^2 - 9 is a difference of squares, which can be factored as:
(x - 3)(x + 3)
So, the final factored form is (n + 1)(x - 3)(x + 3).
By following these steps, you can factor each of these expressions.