PLEASE HELP FACTOR:
1. 8x^4 - 14x^2y^2 + 3y^4
2. 10m^2 - 17mn - 6n^2
3(x+6x)
To factor the given expressions, we'll use techniques such as factoring by grouping, factoring special products, and factoring trinomials.
1. 8x^4 - 14x^2y^2 + 3y^4:
To factor this expression, let's first notice that it resembles a perfect square trinomial with two terms: (a^2 - 2ab + b^2). We can rewrite the expression as:
(4x^2)^2 - 2(4x^2)(y^2) + (y^2)^2
Now, we can factor this as a perfect square trinomial:
(4x^2 - y^2)^2
However, this expression is already in its simplest form, so we cannot factor it further.
2. 10m^2 - 17mn - 6n^2:
To factor this expression, we need to find two numbers that multiply to give the product of the coefficient of the squared term (10) multiplied by the constant term (-6), and add up to give the coefficient of the middle term (-17). In this case, the numbers -20 and 3 satisfy these conditions, since -20 * 3 = -60 and -20 + 3 = -17.
Now, we can decompose the middle term (-17mn) using these two numbers:
10m^2 - 20mn + 3mn - 6n^2
Next, we group the terms in pairs and factor them out:
(10m^2 -20mn) + (3mn - 6n^2)
Factor out the greatest common factor from each pair:
10m(m - 2n) + 3n(m - 2n)
Now, we can factor out the common binomial term, (m - 2n):
(m - 2n)(10m + 3n)
So, the factored form of the expression 10m^2 - 17mn - 6n^2 is (m - 2n)(10m + 3n).