SOLUTION: 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

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: 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      Log On


   

Question 1200924: 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) =
Answer by ikleyn(52794) About Me  (Show Source):
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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

    %282x%5E3+%2B+7x%5E2+%2B+4x+-+4%29%2F%28x%2B2%29 = 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 = %28x%2B2%29%5E2%2A%282x-1%29.


It has the roots x= -2 of multiplicity 2 and x= 1/2 of multiplicity 1.

Solved.