Find the Roots (Zeros) f(x)=2x^3+x^2-7x+1
The given problem is a mathematical question that requires finding the values of x, for which the function f(x) = 2x^3 + x^2 - 7x + 1 equals zero. These values of x are known as the roots or zeros of the cubic polynomial. Essentially, you are asked to determine the x-intercepts of the curve represented by the function, which are the points where the curve crosses the x-axis on a graph.
$f \left(\right. x \left.\right) = 2 x^{3} + x^{2} - 7 x + 1$
Answer
Solution:
Step 1:
Initiate the process by equating the polynomial $f(x) = 2x^{3} + x^{2} - 7x + 1$ to zero.
$$2x^{3} + x^{2} - 7x + 1 = 0$$
Step 2:
Plot the function $y = 2x^{3} + x^{2} - 7x + 1$ and identify the x-coordinates where the graph intersects the x-axis.
The roots are approximately $x \approx -2.19681839, 0.14684217, 1.54997622$.
Step 3:
The x-values obtained from the graph are the approximate solutions to the equation.
Knowledge Notes:
To find the roots (also known as zeros) of the polynomial function $f(x) = 2x^{3} + x^{2} - 7x + 1$, one must understand the following concepts:
Roots of a Polynomial: The roots or zeros of a polynomial are the values of $x$ for which the polynomial equals zero. In other words, these are the solutions to the equation $f(x) = 0$.
Graphing Polynomials: By graphing the polynomial function, one can visually identify the points where the graph crosses the x-axis. These points correspond to the roots of the polynomial.
Approximation: Often, the roots of cubic polynomials cannot be found algebraically, or the algebraic solutions are complex. In such cases, numerical methods or graphing calculators are used to approximate the roots.
Cubic Polynomial: A cubic polynomial is a polynomial of degree three. It has the general form $ax^{3} + bx^{2} + cx + d$, where $a$, $b$, $c$, and $d$ are constants, and $a \neq 0$.
Intersection with the X-axis: When graphing a function, the x-axis represents the line where the value of the function $y$ is zero. Therefore, the points where the graph intersects the x-axis are the solutions to the equation $f(x) = 0$.
In this problem, graphing is used to approximate the roots of the cubic polynomial. The exact algebraic solutions can be more challenging to find, especially if they are not rational numbers. Graphing provides a visual representation and an approximate value for the roots.
