factor the polynomials
3n(squared)+8n=4
8x(to the third) - 6x(squared) - 27x - 54
Both expressions do not have rational factors.
The first (equation) has two real roots, while the second when equated to zero gives one real and two complex roots.
this equation is none factorable
To factor the polynomials, we need to find the common factors of each term and then group them accordingly.
Let's start with the first polynomial: 3n^2 + 8n = 4.
First, move all the terms to one side of the equation to set it equal to zero:
3n^2 + 8n - 4 = 0.
Next, let's try to find the factors of the coefficient of n^2 (which is 3) and the constant term (which is -4). The factors will give us the possible combinations to split the middle term, 8n.
The factors of 3 are 1 and 3, and the factors of -4 are -1, -2, 1, and 2.
Let's try different combinations to find a pair that adds up to 8:
(1n + 4)(3n - 1) = 0.
Therefore, the factored form of the polynomial is:
(1n + 4)(3n - 1) = 0.
Now, let's move on to the second polynomial: 8x^3 - 6x^2 - 27x - 54.
Again, move all the terms to one side of the equation to set it equal to zero:
8x^3 - 6x^2 - 27x - 54 = 0.
Let's look for common factors first. We can see that each term is divisible by 2:
2(4x^3 - 3x^2 - 13.5x - 27) = 0.
Now, we'll try to factor the polynomial 4x^3 - 3x^2 - 13.5x - 27.
Let's look for a rational root by applying the rational root theorem. The possible rational roots are given by the factors of the constant term (which is -27) divided by the factors of the leading coefficient (which is 4).
The factors of 27 are 1, 3, 9, and 27, and the factors of 4 are 1 and 2.
Let's try different combinations to find the rational roots: ±1, ±3, ±9, ±27, ±1/2, ±3/2, ±9/2, and ±27/2.
By evaluating these possibilities, we find that x = -3/2 is a root of the polynomial.
Once we find a root, we can use long division or synthetic division to divide the polynomial by (x + 3/2).
Using synthetic division, we have:
-3/2 | 4 -3 -13.5 -27
| -6 9 27
-------------
4 -9 -4.5 0
Therefore, the factored form of the polynomial is:
2(x + 3/2)(2x^2 - 9x - 4.5) = 0.
Please note that this is just one possible factoring of the polynomial, and further simplification might be possible depending on the context of the problem.