How to Find the Real and Imaginary Parts of a Polynomial

If you want a calculator to split a polynomial into its real and imaginary parts, the key is to tell it what the complex input means. Write the input as $z=x+iy$, where $x$ and $y$ are real. Then expand $p(x+iy)$ and collect the real terms separately from the terms multiplied by $i$.

The result has the form

$$p(x+iy)=u(x,y)+i v(x,y),$$

where $u$ is the real part and $v$ is the imaginary part. A computer algebra system such as SymPy can do the expansion and separation symbolically. The same substitution-and-collecting method works by hand for small polynomials.

What “real and complex parts” usually means

For a complex number $w=a+ib$, its real part is $a$ and its imaginary part is $b$. The imaginary part is the real-valued coefficient of $i$; it is not $ib$ itself. For example, if $w=3-2i$, then $\operatorname{Re}(w)=3$ and $\operatorname{Im}(w)=-2$.

For a polynomial in a complex variable, the parts generally depend on both coordinates of the input. If $p(z)$ has real coefficients, substituting $z=x+iy$ still usually produces both a real part and an imaginary part. If the polynomial has complex coefficients, split those coefficients too as part of the expansion.

A worked example

Take

$$p(z)=z^3+(2-i)z+1$$

and let $z=x+iy$. First expand the cubic:

$$(x+iy)^3=(x^3-3xy^2)+i(3x^2y-y^3).$$

Next expand the linear term:

$$(2-i)(x+iy)=2x+y+i(2y-x).$$

Adding the constant $1$ and grouping the terms gives

$$p(x+iy)=\underbrace{x^3-3xy^2+2x+y+1}_{u(x,y)} +i\underbrace{(3x^2y-y^3+2y-x)}_{v(x,y)}.$$

Therefore,

$$\operatorname{Re}(p(x+iy))=x^3-3xy^2+2x+y+1,$$

and

$$\operatorname{Im}(p(x+iy))=3x^2y-y^3+2y-x.$$

For a numerical check, choose $z=1+2i$, so $x=1$ and $y=2$. The formulas give $u=-6$ and $v=1$, hence $p(1+2i)=-6+i$. Direct evaluation gives the same result.

Use SymPy as a symbolic calculator

In Python with SymPy installed, define $x$ and $y$ as real symbols. That assumption matters: without it, the system may not be able to simplify real-part and imaginary-part expressions fully.

from sympy import symbols, I, expand, re, im

x, y = symbols('x y', real=True)
z = x + I*y
p = z**3 + (2 - I)*z + 1

expanded = expand(p)
real_part = expand(re(expanded))
imaginary_part = expand(im(expanded))

print(real_part)
print(imaginary_part)

The output is

x**3 - 3*x*y**2 + 2*x + y + 1
3*x**2*y - y**3 - x + 2*y

The order of terms may differ, but the expressions are equivalent. For a different polynomial, replace the definition of p; keep x and y declared real and use z = x + I*y.

Common mistakes

• Calling the coefficient of $i$ the imaginary part incorrectly: For $a+ib$, the imaginary part is $b$, not $ib$.
• Forgetting to declare $x$ and $y$ real: A symbolic system may retain unevaluated expressions involving real and imaginary-part operators if it does not know the variables are real.
• Treating a complex coefficient as real: For example, $2-i$ contributes to both parts after multiplication by $x+iy$.
• Confusing the polynomial’s coefficients with its output: A polynomial with real coefficients can still have a nonzero imaginary part when evaluated at a non-real input.

If the input is a specific complex number rather than $x+iy$, substitute its real and imaginary coordinates into $u$ and $v$, or evaluate the polynomial directly and read off the two components.