We want to find the zeros of this polynomial:
[tex]p(x)=\left(2 x^2-9 x+7\right)(x-2)[/tex]
Plot all the zeros ([tex]x[/tex]-intercepts) of the polynomial in the interactive graph.
Answer (1)
Factor the quadratic 2 x 2 − 9 x + 7 into ( 2 x − 7 ) ( x − 1 ) .
Solve ( 2 x − 7 ) ( x − 1 ) = 0 to find roots x = 1 and x = 3.5 .
Solve x − 2 = 0 to find the root x = 2 .
The zeros of the polynomial are 1 , 2 , 3.5 .
Explanation
Understanding the Problem We are given the polynomial p ( x ) = ( 2 x 2 − 9 x + 7 ) ( x − 2 ) and we want to find its zeros, which are the x -values for which p ( x ) = 0 . These are also the x -intercepts of the graph of the polynomial.
Setting up the Equation To find the zeros, we need to solve the equation ( 2 x 2 − 9 x + 7 ) ( x − 2 ) = 0 . This equation is satisfied if either 2 x 2 − 9 x + 7 = 0 or x − 2 = 0 .
Solving the Quadratic Equation First, let's solve the quadratic equation 2 x 2 − 9 x + 7 = 0 . We can try to factor this quadratic. We are looking for two numbers that multiply to 2 × 7 = 14 and add up to − 9 . These numbers are − 2 and − 7 . So we can rewrite the quadratic as 2 x 2 − 2 x − 7 x + 7 = 0 . Factoring by grouping, we get 2 x ( x − 1 ) − 7 ( x − 1 ) = 0 , which simplifies to ( 2 x − 7 ) ( x − 1 ) = 0 . Thus, the solutions to this quadratic equation are 2 x − 7 = 0 \t e x t or x − 1 = 0 , which gives us x = \t f r a c 7 2 = 3.5 and x = 1 .
Solving the Linear Equation Next, let's solve the linear equation x − 2 = 0 . This gives us x = 2 .
Finding the Zeros Therefore, the zeros of the polynomial p ( x ) = ( 2 x 2 − 9 x + 7 ) ( x − 2 ) are x = 1 , x = 2 , and x = 3.5 .
Examples
Understanding the zeros of a polynomial can help in various real-world applications. For instance, in engineering, zeros can represent the stable states of a system. In economics, they can represent equilibrium points where supply equals demand. In physics, they can represent points of zero potential energy. By finding the zeros, we can analyze and predict the behavior of these systems.
