Use the function fx x5 x 2 to complete the following prob

Use the function f(x) = -x^5 + x + 2 to complete the following problems. The leading coefficient is______. The degree of this polynomial function is________. Describe the end behavior of this function\'s graph. Use the rational zero theorem to list the possible rational zeros of f. Use synthetic division with the remainder and factor theorems to find all real squareroots of f.

Solution

22. The Leading coefficient is -1. This is the coefficient of x^5

23. The Degree of polynomial is 5

24. End begavior of graph is linear

25. Rational Zero theorem

       The Possible rational factors are given by p/q where

      q is the integer factors of leading coefficient of polynomial and

       p is the integer factors of constant

      The Polynomial is –x^5 – 2x^4 + x + 2

      Factors of leading coefficient -1 are -1,1

      Factors of constant 2 are 1, 2,-1,-2

     The possible factors are (p/q)

     -1,-2,1,2

26. Synthetic Division

To divide the polynomial by x – 1 use the synthetic division as follows

        1      -1       -2            0              0              1              2

                                -1            -3            -3            -3            -2

-1       -3            -3            -3            -2            0         Remainder

      Since reminder is 0 one of the rational factors is (x-1)

     -x^5 – 2x^4 + x + 2 = (x – 1)(-x^4 – 3x^3 – 3x^2 - 3x - 2)

      Dividing -x^4 – 3x^3 – 3x^2 - 3x – 2 with (x + 2)

-2            -1       -3            -3            -3            -2

   2              2              2              2

-1       -1            -1            -1            0       Remainder

-x^5 – 2x^4 + x + 2 = (x – 1)(X + 2)(-x^3 –x^2 –x -1)

Dividing   -x^3 –x^2 –x -1 b (x + 1)

-1           -1       -1            -1            -1

                                1              0              1

  -1       0              -1            0       Remainder   

-x^5 – 2x^4 + x + 2 = (x – 1)(X + 2)(x+1)(-x2 – 1)

Since –x2 – 1 cannot be factorized the real roots are

1, -2, -1