Find the Roots (Zeros) P(x)=2x^4+2x^3+2x^2+15x-9
The problem you've presented is to determine the values of the variable x for which the polynomial function P(x) equals zero. In mathematical terms, it is asking to solve the equation P(x) = 2x^4 + 2x^3 + 2x^2 + 15x - 9 = 0 for x. Finding the roots (also known as zeros or solutions) of a polynomial means figuring out which values of x make the polynomial evaluate to zero. This is a common task in algebra where one seeks the x-intercepts of the function's graph.
$P \left(\right. x \left.\right) = 2 x^{4} + 2 x^{3} + 2 x^{2} + 15 x - 9$
Answer
Solution:
Step 1:
Initiate the process by equating the polynomial to zero: $2x^{4} + 2x^{3} + 2x^{2} + 15x - 9 = 0$.
Step 2:
Plot the graph of the polynomial function. The roots are where the graph intersects the x-axis. Approximate roots are: $x \approx -2.32111518, 0.53163097$.
Step 3:
No further steps are provided in the original solution. The roots obtained from the graph are the final answer.
Knowledge Notes:
To find the roots of a polynomial, one can follow these steps:
Setting the Polynomial Equal to Zero:
The fundamental theorem of algebra states that a polynomial of degree $n$ has $n$ roots, some of which may be complex or repeated.
To find the roots, we set the polynomial equal to zero: $P(x) = 0$.
Graphing the Polynomial:
Graphing can provide a visual representation of where the function crosses the x-axis, which corresponds to the roots of the polynomial.
Today, graphing can be done using various tools such as graphing calculators or computer software.
Analytical Methods:
For polynomials of degree 4 or lower, there are formulas to find the exact roots, but they can be complex and are not always practical for higher-degree polynomials.
Factoring can sometimes simplify the process if the polynomial can be broken down into simpler binomial or trinomial factors.
Numerical Methods:
When analytical methods are not feasible, numerical methods such as the Newton-Raphson method or synthetic division can be used to approximate the roots.
These methods involve iteration and can provide approximate values for the roots.
Complex Roots:
If a polynomial has real coefficients, any complex roots will occur in conjugate pairs.
This means that if $(a + bi)$ is a root, then $(a - bi)$ will also be a root.
In the given problem, the solution involves graphing the polynomial to find the approximate values of the roots. The exact analytical solution is not provided, which suggests that the roots may not be easily factorable or that the solution is focused on a numerical approximation.
