1) 27x^3+8
2) 3x^2-x-4
The first is the sum of two cubes, you have a formula for that.
The second:
(3x -4)(x +1)
a^3 + b^3 = (a+b)(a^2-ab+b^2)
here
a = 3x
b = 2
You take it from there
(3x-4)(x+1)
1) To factor the expression 27x^3 + 8, we can use the sum and difference of cubes identity. The sum of cubes identity states that a^3 + b^3 can be factored as (a + b)(a^2 - ab + b^2), and the difference of cubes identity states that a^3 - b^3 can be factored as (a - b)(a^2 + ab + b^2).
In our expression, 27x^3 + 8 can be seen as (3x)^3 + 2^3. Recognizing the expression as the sum of cubes, we can write it as (3x + 2)((3x)^2 - (3x)(2) + 2^2). Simplifying further, we get (3x + 2)(9x^2 - 6x + 4).
So, the factored form of 27x^3 + 8 is (3x + 2)(9x^2 - 6x + 4).
2) To factor the expression 3x^2 - x - 4, we can use the product-sum method or quadratic factoring.
First, we need to find two numbers that multiply to -12 (the product of the coefficient of x^2 and the constant term) and add up to -1 (the coefficient of the x term). In this case, the numbers are -4 and 3.
Next, we rewrite the expression using these numbers: 3x^2 - 4x + 3x - 4.
Now, we group the terms and factor by grouping: (3x^2 - 4x) + (3x - 4).
Taking out the common factors from each group, we get x(3x - 4) + 1(3x - 4).
Finally, we can see that (3x - 4) is common to both terms. Factoring it out, we have (3x - 4)(x + 1).
So, the factored form of 3x^2 - x - 4 is (3x - 4)(x + 1).