We want to find the zeros of this polynomial:
[tex]p(x)=(x+2)(2 x-3)(x-3)[/tex]
Plot all the zeros ($x$-intercepts) of the polynomial in the interactive graph.
Answer (2)
Set the polynomial p ( x ) to zero: ( x + 2 ) ( 2 x − 3 ) ( x − 3 ) = 0 .
Solve for x in each factor: x + 2 = 0 , 2 x − 3 = 0 , x − 3 = 0 .
Find the roots: x = − 2 , x = 2 3 = 1.5 , x = 3 .
The zeros of the polynomial are: − 2 , 1.5 , 3 .
Explanation
Understanding the Problem We are given the polynomial p ( x ) = ( x + 2 ) ( 2 x − 3 ) ( x − 3 ) and we want to find its zeros, which are the x -values for which p ( x ) = 0 . These values are also the x -intercepts of the graph of the polynomial.
Setting the Polynomial to Zero To find the zeros, we set p ( x ) = 0 and solve for x :
( x + 2 ) ( 2 x − 3 ) ( x − 3 ) = 0
Finding the Roots This equation is satisfied if any of the factors are equal to zero. Thus, we have three possible cases:
Case 1: x + 2 = 0 Case 2: 2 x − 3 = 0 Case 3: x − 3 = 0
Solving for x Now we solve each case for x :
Case 1: x + 2 = 0 ⟹ x = − 2 Case 2: 2 x − 3 = 0 ⟹ 2 x = 3 ⟹ x = 2 3 = 1.5 Case 3: x − 3 = 0 ⟹ x = 3
The Zeros of the Polynomial Therefore, the zeros of the polynomial are x = − 2 , x = 1.5 , and x = 3 . 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 polynomial can represent the stable states of a system. In economics, they can represent equilibrium points in a market. In physics, they can represent the points where a potential energy function is minimized. By finding the zeros, we can analyze and predict the behavior of these systems.
The zeros of the polynomial p ( x ) = ( x + 2 ) ( 2 x − 3 ) ( x − 3 ) are found by setting the polynomial equal to zero and solving for x. The solutions are x = − 2 , x = 1.5 , and x = 3 . These values represent the x-intercepts of the polynomial's graph.
;
