Answer
The rational zeros are $x = -1$ or $x = 5$ or $x = 2$ or $x = -\frac{1}{3}$.
The polynomial in factored form is $(x - 5) (x - 2) (x + 1)^2 (3 x + 1)$.
Work Step by Step
Alternate form:
$3x^5-14x^4-14x^3+36x^2+43x+10 = (x + 1)^2 (3 x^3 - 20 x^2 + 23 x + 10)$
Solve for x over the real numbers:
$(x + 1)^2 (3 x^3 - 20 x^2 + 23 x + 10) = 0$
Split into two equations:
$(x + 1)^2 = 0$ or $3 x^3 - 20 x^2 + 23 x + 10 = 0$
Take the square root of both sides:
$x + 1 = 0$ or $3 x^3 - 20 x^2 + 23 x + 10 = 0$
Subtract 1 from both sides:
$x = -1$ or $3 x^3 - 20 x^2 + 23 x + 10 = 0$
The left hand side factors into a product with three terms:
$x = -1$ or $(x - 5) (x - 2) (3 x + 1) = 0$
Split into three equations:
$x = -1$ or $x - 5 = 0$ or $x - 2 = 0$ or $3 x + 1 = 0$
Add 5 to both sides:
$x = -1$ or $x = 5$ or $x - 2 = 0$ or $3 x + 1 = 0$
Add 2 to both sides:
$x = -1$ or $x = 5$ or $x = 2$ or $3 x + 1 = 0$
Subtract 1 from both sides:
$x = -1$ or $x = 5$ or $x = 2$ or $3 x = -1$
Answer:
The rational zeros are $x = -1$ or $x = 5$ or $x = 2$ or $x = -\frac{1}{3}$.
The polynomial in factored form is $(x - 5) (x - 2) (x + 1)^2 (3 x + 1)$.
