I posted this earlier in the day but no one answered. Can anyone help with these problems?
Factor completely:
1. 8x3 + 2x3 – 12x –3
2. 27x3+8
3.125 – 8x3
The second one is the sum of perfect cubes, for which you have a formula.
The third is the difference of perfect cubes, same formula almost.
The first I assume is 8x^3+2x^2-12x-3=0 This is not going to factor nicely, I suggest you graph it on your calculator, and take a look at the roots. Not a nice function.
I don't have a graphing calculator. Is there another way to find the roots so I can factor the problem? And I'm sorry I don't know what formula you are talking about for 1 and 2.
I can definitely help you with these problems. Let me guide you through the process of factoring each of them.
1. To factor 8x^3 + 2x^3 - 12x - 3:
First, combine like terms: 8x^3 + 2x^3 - 12x - 3 = 10x^3 - 12x - 3.
Now, we need to find common factors among the terms. In this case, there aren't any common factors other than 1.
Next, we look for any possible rational roots. Potential rational roots can be found using the rational root theorem, which states that a rational root of a polynomial equation can be expressed as p/q, where p is a factor of the constant term (in this case, -3) and q is a factor of the leading coefficient (in this case, 10). In this case, the potential rational roots are ±1, ±3, ±1/2, and ±3/2.
By testing these potential roots using synthetic division or substituting them into the polynomial, we find that none of them are roots of 10x^3 - 12x - 3. Therefore, the polynomial is prime, meaning it cannot be factored further.
2. To factor 27x^3 + 8:
This expression is a sum of two cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2).
In this case, we have 27x^3 + 8 = (3x)^3 + 2^3.
Using the sum of cubes formula, we can write it as (3x + 2)((3x)^2 - (3x)(2) + 2^2).
Simplifying further, we get (3x + 2)(9x^2 - 6x + 4), which is the fully factored form.
3. To factor 125 - 8x^3:
Similar to the previous problem, this expression is a difference of cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2).
Here, we have 125 - 8x^3 = 5^3 - (2x)^3.
Using the difference of cubes formula, we can write it as (5 - 2x)((5)^2 + (5)(2x) + (2x)^2).
Simplifying further, we get (5 - 2x)(25 + 10x + 4x^2), which is the fully factored form.
I hope this explanation helps you understand how to factor these polynomials. If you have any further questions, feel free to ask!