Numbers and Polynomials Reference
Complex Numbers
Complex numbers extend the real number system so every quadratic equation has solutions.
Solved Examples
Jump directly to a worked example by operation or equation type.
Fact Table
| Idea | Rule | Watch For |
|---|---|---|
| Imaginary unit | $i^2=-1$. | Replace every $i^2$ with $-1$. |
| Standard form | $a+bi$, where $a$ and $b$ are real. | Combine real parts with real parts and imaginary parts with imaginary parts. |
| Conjugates | $a+bi$ and $a-bi$. | Their product is the real number $a^2+b^2$. |
| Powers of $i$ | $i, -1, -i, 1$ repeats every four powers. | Reduce the exponent modulo $4$. |
Content Formulas
Product
$$(a+bi)(c+di)=(ac-bd)+(ad+bc)i$$
Conjugate Product
$$(a+bi)(a-bi)=a^2+b^2$$
Quadratic Formula
$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
Classic Examples
Add Complex Numbers
Simplify $(3+2i)+(4-5i)$.
Solution Steps
- Group the real parts and the imaginary parts.
- Add each pair.
- State the result in standard form.
$$(3+4)+(2i-5i)$$$$7-3i$$$$\boxed{7-3i}$$
Multiply Complex Numbers
Simplify $(2+i)(3-4i)$.
Solution Steps
- Distribute each term in the first factor.
- Replace $i^2$ with $-1$.
- Combine the real and imaginary parts.
$$6-8i+3i-4i^2$$$$6-5i+4$$$$\boxed{10-5i}$$
Simplify a Power of $i$
Simplify $i^{37}$.
Solution Steps
- Divide the exponent by the cycle length $4$.
- Use the remainder to identify the matching power.
- State the simplified value.
$$37=4(9)+1$$$$i^{37}=\left(i^4\right)^9i$$$$\boxed{i^{37}=i}$$
Solve a Quadratic with Nonreal Roots
Solve $x^2+4x+13=0$.
Solution Steps
- Use the quadratic formula with $a=1$, $b=4$, and $c=13$.
- Simplify the negative discriminant using $\sqrt{-36}=6i$.
- Reduce the fractions and state both roots.
$$x=\frac{-4\pm\sqrt{4^2-4(1)(13)}}{2}$$$$x=\frac{-4\pm\sqrt{-36}}{2}=\frac{-4\pm6i}{2}$$$$\boxed{x=-2\pm3i}$$