Answer
The real zeros are $x = 2$ or $x = 3$ or $x = -1$ or $x = -2$.
The polynomial in factored form is $(x - 3) (x - 2)^2 (x + 1) (x + 2)$.
Work Step by Step
Alternate form:
$x^5 - 4 x^4 - 3 x^3 + 22 x^2 - 4 x - 24 = (x - 2)^2 (x^3 - 7 x - 6)$
Solve for x over the real numbers:
$(x - 2)^2 (x^3 - 7 x - 6) = 0$
Split into two equations:
$(x - 2)^2 = 0$ or $x^3 - 7 x - 6 = 0$
Take the square root of both sides:
$x - 2 = 0$ or $x^3 - 7 x - 6 = 0$
Add 2 to both sides:
$x = 2$ or $x^3 - 7 x - 6 = 0$
The left hand side factors into a product with three terms:
$x = 2$ or $(x - 3) (x + 1) (x + 2) = 0$
Split into three equations:
$x = 2$ or $x - 3 = 0$ or $x + 1 = 0$ or $x + 2 = 0$
Add 3 to both sides:
$x = 2$ or $x = 3$ or $x + 1 = 0$ or $x + 2 = 0$
Subtract 1 from both sides:
$x = 2$ or $x = 3$ or $x = -1$ or $x + 2 = 0$
Answer:
The real zeros are $x = 2$ or $x = 3$ or $x = -1$ or $x = -2$.
The polynomial in factored form is $(x - 3) (x - 2)^2 (x + 1) (x + 2)$.
