For the following find the function P defined by a polynomia

For the following, find the function P defined by a polynomial of degree 3 with real coefficients that satisfy the given conditions. The zeros are - 3, - 1, and 4. (Simplify your answer.)

Solution

Let the required polynomial be p(x) = a (x+ 3) (x + 1) ( x -4) ( The factors have been chosen in accordance with the zeros of the polynomial p(x)).

Since p ( -2 ) = -30, on substituting x = -2 in the expression forp ( x ) as above, we get a ( -2 + 3) ( -2 + 1) ( -2 - 4) = -30 or, 1*(-1)*(-6) = -30 or, 6a = -30 so that a = -5. Then p(x) = - 5 (x+ 3) (x + 1) ( x -4) = -5( x² + 4x + 3) ( x -4) = -5( x³ + 4x² + 3x - 4x² - 16x -12) = -5( x³ -13x -12) = -5x³ + 65 x + 60. We can verify the result by substituting x = -3, -1 and 4 in p(x). Each time, we get 0.Thus, the required polynomial is p(x) =  -5x³ + 65 x + 60