Question

Hii guys please answer this question

Answer 1

Answer:

  -1/2

Step-by-step explanation:

Zeros of a polynomial function are its x-intercepts, the values of x that make the polynomial zero. Each zero x=p corresponds to a linear factor (x-p) of the polynomial.

Monic polynomial

A monic polynomial is one that has a leading coefficient of 1. For a monic polynomial of degree n, the coefficient of the term of degree n-1 is the opposite of the sum of all of the zeros. The constant term of any odd-degree monic polynomial is the opposite of the product of the zeros.

The given polynomial can be made monic by dividing by its leading coefficient:

  (2x^3 -x^2 -5x -2)/2 = x^3 -(1/2)x^2 -(5/2)x -1

Sum of zeros

The given polynomial has degree 3, so we're interested in the coefficient of the x^2 term. That coefficient is -1/2, so the sum of zeros will be ...

  sum of zeros = -(-1/2) = 1/2

If z represents the third zero, then we have the sum ...

  1/2 = -1 +2 +z

  z = -1/2 . . . . . . subtract 1 from both sides

The third zero is -1/2.

Product of zeros

The given polynomial is of odd degree, so the product of the zeros is the opposite of the constant.

  (-1)(2)(z) = -(-1)

  z = 1/(-2) = -1/2 . . . . . divide by the coefficient of z

The third zero is -1/2.

Graph

Of course, a graph will show all of the (real) zeros. The attached graph from a graphing calculator shows the third zero is -1/2.