Factor the following polynomial.
x^3+5x^2+6x+30
To factor the polynomial x^3+5x^2+6x+30, we first look for any common factors among the coefficients. In this case, there are no common factors other than 1.
Next, we consider possible rational roots of the polynomial. A rational root is a number of the form p/q, where p is a factor of the constant term (30) and q is a factor of the leading coefficient (1). The factors of 30 are ±1, ±2, ±3, ±5, ±6, ±10, ±15, and ±30, and the factors of 1 are ±1. Therefore, the possible rational roots are ±1, ±2, ±3, ±5, ±6, ±10, ±15, and ±30.
Using synthetic division or substitution, we find that none of these possible rational roots are actual roots of the polynomial. Therefore, the polynomial cannot be factored using rational roots.
Alternatively, we can try to factor by grouping. Let's rearrange the terms of the polynomial:
x^3 + 6x + 5x^2 + 30
Now, let's group the terms as follows:
(x^3 + 6x) + (5x^2 + 30)
Factoring out the greatest common factor from each group gives us:
x(x^2 + 6) + 5(x^2 + 6)
Notice that we now have a common factor of (x^2 + 6) in both terms. Factor it out:
(x^2 + 6)(x + 5)
Therefore, the factored form of the polynomial x^3+5x^2+6x+30 is (x^2 + 6)(x + 5).