Answer
The rational zeros are $x=2$ or $x=-1$ or $x=\frac{1}{2}$.
The polynomial in factored form is $(x-2)^2(x+1)(2x-1)$.
Work Step by Step
Alternate form:
$2x^4-7x^3+3x^2+8x-4 = (x-2)^2(2x^2+x-1)$
Solve for $x$ over the real numbers:
$(x-2)^2(2x^2+x-1) = 0$
Split into two equations:
$(x-2)^2=0$ or $2x^2+x-1=0$
Take the square root of both sides:
$x-2=0$ or $2x^2+x-1=0$
Add $2$ to both sides:
$x=2$ or $2x^2+x-1=0$
The left hand side factors into a product with two terms:
$x=2$ or $(2x-1)(x+1)=0$
Split into two equations:
$x=2$ or $x+1=0$ or $2x-1=0$
Subtract $1$ from both sides:
$x=2$ or $x=-1$ or $2x-1=0$
Add $1$ to both sides:
$x=2$ or $x=-1$ or $2x=1$
Answer:
The rational zeros are $x=2$ or $x=-1$ or $x=\frac{1}{2}$.
The polynomial in factored form is $(x-2)^2(x+1)(2x-1)$.
