A polynomial *p(x)* is an expression of the form:

\(p(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + … + a_nx^n\)

Where *n* is any non-negative integer.

## Solve a polynomial p(x) in R

To solve the equation \(p(x) = 0\) in R, we can use the function: *polyroot*.

For example, let’s solve the equation:

\(p(x) = -\sqrt{2} – x + x^2 + 0.4x^3 = 0\)

# we use the coefficients of p(x) as inputs to polyroot solutions = polyroot(c(-sqrt(2), -1, 1, 0.4)) print(solutions) # outputs: 1.3390011-0i -0.8977101+0i -2.9412910-0i

## Check if the solutions are real numbers

So, we can see that the solutions of *p(x)* are all real numbers (associated with *0i*), but we can check for this in R:

# extracting the imaginary parts of the solutions Im(solutions) # outputs: -3.333339e-18 6.981698e-18 -3.648359e-18 # Which are numbers very close to 0, # To get them to be exactly 0, we need to round our answer: round(Im(solutions), 5) # outputs: 0 0 0 # Finally, the following code outputs: # TRUE if the solutions are real # FALSE otherwise all(round(Im(solutions), 5) == 0) # outputs: TRUE # With the help of this code, # we can remove the imaginary parts of the solutions # if they are real numbers if(all(round(Im(solutions), 5) == 0)) { solutions = Re(solutions) # Re outputs the real part of a complex number } print(solutions) # outputs: 1.3390011 -0.8977101 -2.9412910

## Plot p(x) and its solutions

# Coding the polynomial p(x) in R p = function(x) { -sqrt(2) -x + x^2 + 0.4*x^3 } # choosing the domain of p(x) x = seq(-5, 5, by = 0.1) # so x is: -5, -4.9, -4.8, ..., 4.8, 4.9, 5 # plotting p(x) and limiting its range to -5, 5 plot(x, p(x), type = 'l', col = 'red', ylim = c(-5, 5)) # adding the x-axis abline(h = 0, col = 'blue') # adding the solutions points(x = solutions, y = rep(0, length(solutions)), # x and y coordinates of the solutions pch = 18, cex = 2, col = 'red') # pch controls the shape of the points # annotations solutions = sort(solutions) # to annotate them from small to large text(x = solutions, y = rep(0, length(solutions)), labels = paste('Solution', seq(1, length(solutions))), pos = 3, col = 'red')

**Output:**