We want to find the zeros of this polynomial:
[tex]p(x)=(x-1)(x+3)(2 x+1)[/tex]
Plot all the zeros ([tex]x[/tex]-intercepts) of the polynomial in the interactive graph.
Answer (1)
The zeros of the polynomial p ( x ) = ( x − 1 ) ( x + 3 ) ( 2 x + 1 ) are found by setting each factor to zero and solving for x . The solutions are:
Set x − 1 = 0 , which gives x = 1 .
Set x + 3 = 0 , which gives x = − 3 .
Set 2 x + 1 = 0 , which gives x = − 2 1 .
The zeros are x = − 3 , − 0.5 , 1 .
The zeros of the polynomial are − 3 , − 0.5 , 1 .
Explanation
Understanding the Problem We are given the polynomial p ( x ) = ( x − 1 ) ( x + 3 ) ( 2 x + 1 ) and we want to find its zeros, which are the values of x for which p ( x ) = 0 . These zeros correspond to the x -intercepts of the polynomial's graph.
Setting up the Equation To find the zeros, we set p ( x ) = 0 and solve for x :
( x − 1 ) ( x + 3 ) ( 2 x + 1 ) = 0
Solving for x This equation is satisfied if any of the factors are equal to zero. So we have three cases:
Case 1: x − 1 = 0 Solving for x , we get x = 1 .
Case 2: x + 3 = 0 Solving for x , we get x = − 3 .
Case 3: 2 x + 1 = 0 Solving for x , we get 2 x = − 1 , so x = − 2 1 = − 0.5 .
Listing the Zeros Therefore, the zeros of the polynomial are x = 1 , x = − 3 , and x = − 0.5 . These are the x -intercepts of the graph of the polynomial.
Examples
Understanding the zeros of a polynomial is crucial in many real-world applications. For instance, in engineering, the zeros of a characteristic equation determine the stability of a system. In economics, they can represent equilibrium points in a market. Knowing the zeros helps predict the behavior of the system or model being studied. For example, if p ( x ) represents the profit of a company as a function of the number of products sold ( x ), the zeros of p ( x ) would indicate the break-even points where the company neither makes a profit nor incurs a loss. Finding these points is essential for business planning and decision-making.
