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 values 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 Let's first 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 x = 1 and x = 2 7 = 3.5 .
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 ) are x = 1 , x = 2 , and x = 3.5 . These are the x -intercepts of the polynomial's graph.
Examples
Finding the zeros of a polynomial is a fundamental concept in algebra and calculus. In real life, understanding the zeros of a function can help in various applications. For example, if you are designing a bridge, you might model the load on the bridge as a polynomial function. The zeros of this function would represent the points where the load is zero, which are important for understanding the structural integrity of the bridge. Similarly, in business, the zeros of a profit function can represent the break-even points, where the company neither makes nor loses money. Knowing these points is crucial for making informed business decisions. The zeros of the polynomial p ( x ) = ( 2 x 2 − 9 x + 7 ) ( x − 2 ) are 1, 2, and 3.5.
