.
Find all rational zeros of the polynomial. (Enter your answers as a comma-separated list. Enter all answers including repetitions.)
P(x) = 2x^3 + 7x^2 + 4x − 4
x =
Write the polynomial in factored form.
P(x) =
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
According to Rational Zeroes theorem, the list of possible rational zeroes consists of these values
{1, -1, 2, -2, 4, -4}.
It is easy to check that x= -2 is the root.
Then the given polynomial is divisible by (x+2), so we divide the given polynomial
by (x+2) to reduce the degree
= 2x^2 + 3x - 2.
Regarding quadratic polynomial 2x^2 + 3x - 2, we can factor it by grouping
2x^2 + 3x - 2 = (2x^2 + 4x) - (x+2) = 2x*(x+2) - (x+2) = (2x-1)*(x+2).
Therefore, the final decomposition of the given polynomial is
2x^3 + 7x^2 + 4x − 4 = .
It has the roots x= -2 of multiplicity 2 and x= 1/2 of multiplicity 1.
Solved.