We want to find the zeros of this polynomial:
[tex]p(x)=(x+2)(2 x+7)(x-1)(x-3)[/tex]
Plot all the zeros ( [tex]x[/tex]-intercepts) of the polynomial in the interactive graph.
Answer (1)
Set the polynomial p ( x ) to zero: ( x + 2 ) ( 2 x + 7 ) ( x − 1 ) ( x − 3 ) = 0 .
Solve each factor for x to find the zeros.
The zeros are x = − 2 , x = − 2 7 , x = 1 , and x = 3 .
The zeros of the polynomial are − 3.5 , − 2 , 1 , 3 .
Explanation
Understanding the Problem We are given the polynomial p ( x ) = ( x + 2 ) ( 2 x + 7 ) ( x − 1 ) ( x − 3 ) and we want to find its zeros. The zeros of a polynomial are the values of x for which p ( x ) = 0 . In other words, we are looking for the x -intercepts of the graph of the polynomial.
Setting the Polynomial to Zero To find the zeros, we set p ( x ) = 0 :
( x + 2 ) ( 2 x + 7 ) ( x − 1 ) ( x − 3 ) = 0
Solving for x Since the polynomial is already factored, we can find the zeros by setting each factor equal to zero and solving for x :
x + 2 = 0 ⟹ x = − 2
2 x + 7 = 0 ⟹ 2 x = − 7 ⟹ x = − 2 7 = − 3.5
x − 1 = 0 ⟹ x = 1
x − 3 = 0 ⟹ x = 3
Finding the Zeros Therefore, the zeros of the polynomial are x = − 3.5 , − 2 , 1 , 3 . These are the x -intercepts of the graph of the polynomial.
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 understand the behavior of the structure under different loads. This often involves finding the roots of a polynomial equation that describes the structure's response to the load. Similarly, in physics, finding the zeros of a polynomial can help determine the equilibrium points of a system. Knowing the zeros allows us to predict when the system will be stable or unstable. For instance, consider a simple harmonic oscillator whose motion is described by a polynomial equation. The zeros of this polynomial can tell us the points where the oscillator will come to rest or change direction.
