8.1 Polynomial Equations

Let's start by solving a simple quadratic equation, x ² + 3 x ? 5 = 0. Example 8.1.1 shows how to use the function Solve to do this.

In[1]= (*   solving a polynomial equation *)       <b class="bold">Solve[x<sup2</sup> + 3x-5==0, x]</b>

Solve takes two arguments. The first is the equation we want to solve and the second is the variable that we want to solve for. Notice that we have used double equal signs in the equation. Recall that a single equal sign is not used to represent an abstract equation, but rather to replace the left-hand side of the expression with the right-hand side. For example a=2 assigns the value 2 to the variable a.

Mathematica finds that there are two solutions, that is, values of x that make the equation true. In general, any number that makes a polynomial zero is called a root of the polynomial. So in this case, the solutions of the equation are also the roots of the polynomial x ² + 3 x ?5. Instead of just listing the solutions, Mathematica gives replacement rules of the form {x ?x ₀}, where