We want to find the zeros of this polynomial:
[tex]p(x)=2 x^3-x^2-8 x+4[/tex]
Plot all the zeros ( [tex]x[/tex]-intercepts) of the polynomial in the interactive graph.
Answer (1)
The task is to find the zeros of the polynomial p ( x ) = 2 x 3 − x 2 − 8 x + 4 .
Factor the polynomial by grouping: p ( x ) = ( x − 2 ) ( x + 2 ) ( 2 x − 1 ) .
Set each factor to zero and solve for x .
The zeros of the polynomial are x = − 2 , x = 2 1 , x = 2 .
The zeros are − 2 , 2 1 , 2 .
Explanation
Understanding the Problem We are given the polynomial p ( x ) = 2 x 3 − x 2 − 8 x + 4 and we want to find its zeros, which are the x -values where p ( x ) = 0 . These zeros correspond to the x -intercepts of the polynomial's graph.
Factoring the Polynomial To find the zeros, we need to solve the equation 2 x 3 − x 2 − 8 x + 4 = 0 . We can try factoring by grouping.
Factoring by Grouping We group the terms as follows: ( 2 x 3 − x 2 ) + ( − 8 x + 4 ) . From the first group, we can factor out x 2 , and from the second group, we can factor out − 4 . This gives us x 2 ( 2 x − 1 ) − 4 ( 2 x − 1 ) . Now we can factor out the common factor ( 2 x − 1 ) to get ( x 2 − 4 ) ( 2 x − 1 ) .
Difference of Squares We can further factor x 2 − 4 as a difference of squares: x 2 − 4 = ( x − 2 ) ( x + 2 ) . So, the factored form of the polynomial is p ( x ) = ( x − 2 ) ( x + 2 ) ( 2 x − 1 ) .
Finding the Zeros Now we set p ( x ) = 0 to find the zeros: ( x − 2 ) ( x + 2 ) ( 2 x − 1 ) = 0 . This equation is satisfied if any of the factors are equal to zero.
Solving for x We solve each factor for x :
x − 2 = 0 ⇒ x = 2
x + 2 = 0 ⇒ x = − 2
2 x − 1 = 0 ⇒ 2 x = 1 ⇒ x = 2 1
The Zeros Therefore, the zeros of the polynomial are x = − 2 , 2 1 , 2 . 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 engineering and physics. For example, when designing a bridge, engineers need to analyze the polynomial equations that describe the bridge's structure. The zeros of these polynomials can represent critical points where the structure experiences maximum stress or deflection. By identifying these points, engineers can reinforce the structure to ensure its stability and safety. Similarly, in physics, finding the zeros of a polynomial can help determine the equilibrium points of a system or the resonant frequencies of a circuit.
