Question: Find all solutions of the following polynomial equation. 30 + 19x + 6x^3 - 35x^2 = 0 Answer…

Find all solutions of the following polynomial equation. 30 + 19x + 6x^3 - 35x^2 = 0 Answer Separate multiple answers with commas. {

Solution

To solve the polynomial equation \( 30 + 19x + 6x^3 - 35x^2 = 0 \), we first rearrange it in standard form: \[ 6x^3 - 35x^2 + 19x + 30 = 0 \] Next, we will try to find the rational roots using the Rational Root Theorem, which suggests that any rational solution, p/q, is a factor of the constant term (30) divided by a factor of the leading coefficient (6). Possible factors of 30: \( \pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30 \) Possible factors of 6: \( \pm 1, \pm 2, \pm 3, \pm 6 \) Possible rational roots: \( \frac{\pm 1}{\pm 1}, \frac{\pm 1}{\pm 2}, \frac{\pm 1}{\pm 3}, \frac{\pm 1}{\pm 6}, \frac{\pm 2}{\pm 1}, \frac{\pm 2}{\pm 2}, \frac{\pm 2}{\pm 3}, \frac{\pm 2}{\pm 6}, \ldots \) This will result in a list of possible rational roots: \( \pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{5}{2}, \pm \frac{1}{3}, \pm \frac{5}{3}, \pm \frac{10}{3}, \pm \frac{1}{6}, \pm \frac{5}{6} \) Test these possible roots using synthetic division or substitution to see which ones satisfy the equation. Let’s test \( x = 1 \): \[ 6(1)^3 - 35(1)^2 + 19(1) + 30 = 6 - 35 + 19 + 30 = 20 \neq 0 \] Test \( x = -1 \): \[ 6(-1)^3 - 35(-1)^2 + 19(-1) + 30 = -6 - 35 - 19 + 30 = -30 \neq 0 \] Continue this process for other possible rational roots until you find the one for which the polynomial evaluates to zero. Finally, once a root is found, use synthetic division or polynomial division to factor the polynomial and find the remaining roots by solving the resulting polynomial equation. Repeat the process until all roots are found.