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 the quadratic equation are 2 x − 7 = 0 × x = \tIrac 7 2 = 3.5 and x − 1 = 0 × x = 1 .
Solving the Linear Equation Now, 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 .
Final Answer The zeros of the polynomial are 1 , 2 , and 3.5 . 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 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 signal processing, the zeros of a polynomial can represent frequencies that are filtered out or amplified by a system.
