Use the rational zeros theorem to find all the real zeros of

Use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. f(x) = x^4 + x^3 - 11x^2 - 9x +18 What are the real zeros? Select the correct choice below and, if necessary, fill in the answer box to complete your answer. x = (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any rational numbers in the expression. Use a comma to separate answers as needed.) There are no real zeros. Use the real zeros to factor f. F(x) = (Simplify your answer. Type your answer in factored form. Type an exact answer, using radicals as needed. Use integers or fractions for any rational numbers in the expression.)

Solution

f(x) = x^4 + x^3 - 11x^2 - 9x + 18

by rational root test the polynomial has rational zeros at +- { 1 , 2, 3, 6 , 9 , 18 }

at x = 1 the polynomial has real zero

therefore, dividing f(x) by x-1 we get

x^3+ 2x^2-9x-18

again by rational root test actual zero occur at x = -2

again dividing x^3+2x^2-9x-18 by x+2

we get x^2 - 9

hence solving the quadratic by difference of squares method we get x = 3 , -3

the 4 real zeros are

x = 1

x =-2

x = 3

x = -3