We want to find the zeros of this polynomial:
[tex]p(x)=2 x^3+4 x^2-6 x[/tex]
Plot all the zeros ($x$-intercepts) of the polynomial in the interactive graph.
Answer (1)
Factor out the common factor: p ( x ) = 2 x ( x 2 + 2 x − 3 ) .
Factor the quadratic expression: x 2 + 2 x − 3 = ( x + 3 ) ( x − 1 ) .
Set p ( x ) = 0 and solve for x: 2 x ( x + 3 ) ( x − 1 ) = 0 .
The zeros are x = − 3 , 0 , 1 .
Explanation
Understanding the Problem We are given the polynomial p ( x ) = 2 x 3 + 4 x 2 − 6 x and we want to find its zeros, which are the x -values for which p ( x ) = 0 . These zeros correspond to the x -intercepts of the polynomial's graph.
Factoring the Polynomial First, we can factor out the common factor of 2 x from the polynomial: p ( x ) = 2 x ( x 2 + 2 x − 3 ) Now, we need to factor the quadratic expression x 2 + 2 x − 3 . We are looking for two numbers that multiply to -3 and add to 2. These numbers are 3 and -1. So, we can factor the quadratic as: x 2 + 2 x − 3 = ( x + 3 ) ( x − 1 ) Therefore, the factored form of the polynomial is: p ( x ) = 2 x ( x + 3 ) ( x − 1 )
Finding the Zeros To find the zeros of the polynomial, we set p ( x ) = 0 and solve for x :
2 x ( x + 3 ) ( x − 1 ) = 0 This equation is satisfied if any of the factors are equal to zero. Thus, we have three possible solutions:
2 x = 0 ⟹ x = 0
x + 3 = 0 ⟹ x = − 3
x − 1 = 0 ⟹ x = 1 So, the zeros of the polynomial are x = 0 , x = − 3 , and x = 1 .
The Zeros The zeros of the polynomial p ( x ) = 2 x 3 + 4 x 2 − 6 x are x = − 3 , 0 , 1 . These are the points where the graph of the polynomial intersects the x -axis.
Examples
Understanding the zeros of a polynomial is crucial in many areas, such as physics and engineering. For example, when designing a bridge, engineers need to analyze the forces acting on the structure. These forces can often be modeled by polynomial equations, and finding the zeros of these polynomials helps engineers determine the points where the structure is most vulnerable to stress or failure. Similarly, in physics, understanding the zeros of a polynomial can help predict the behavior of a system, such as the trajectory of a projectile or the stability of an electrical circuit. By finding the zeros, we can determine critical points and make informed decisions about the design and operation of these systems. The zeros of the polynomial p ( x ) = 2 x 3 + 4 x 2 − 6 x are -3, 0, and 1.
