Finding the roots of a polynomial involves determining the values of the variable (usually denoted as \( x \)) that make the polynomial equal to zero. Here are several methods to find roots, along with examples.
1. Factoring
One of the simplest methods for polynomials is factoring. This is particularly effective for quadratic polynomials or simple cubic ones.
Example: Find the roots of \( p(x) = x^2 - 5x + 6 \).
First, we factor the polynomial:
\[ p(x) = x^2 - 5x + 6 = (x - 2)(x - 3) \]
To find the roots, set each factor equal to zero:
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \] \[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]
So the roots are \( x = 2 \) and \( x = 3 \).
2. Using the Quadratic Formula
For any quadratic polynomial of the form \( ax^2 + bx + c \), the roots can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Example: Find the roots of \( p(x) = 2x^2 - 4x - 6 \).
Here, \( a = 2 \), \( b = -4 \), and \( c = -6 \).
Calculate the discriminant:
\[ D = b^2 - 4ac = (-4)^2 - 4 \cdot 2 \cdot (-6) = 16 + 48 = 64 \]
Now apply the quadratic formula:
\[ x = \frac{-(-4) \pm \sqrt{64}}{2 \cdot 2} = \frac{4 \pm 8}{4} \]
This gives:
- \( x = \frac{12}{4} = 3 \)
- \( x = \frac{-4}{4} = -1 \)
So the roots are \( x = 3 \) and \( x = -1 \).
3. Using Synthetic Division or Polynomial Division
For higher-degree polynomials, especially cubics and quartics, we can use synthetic division or polynomial long division.
Example: Find the roots of \( p(x) = x^3 - 6x^2 + 11x - 6 \).
First, try possible rational roots (based on the Rational Root Theorem). The possible rational roots from the factors of -6 are \( \pm 1, \pm 2, \pm 3, \pm 6 \).
Testing \( x = 1 \):
\[ p(1) = 1^3 - 6(1^2) + 11(1) - 6 = 1 - 6 + 11 - 6 = 0 \]
So \( x = 1 \) is a root. We can now divide \( p(x) \) by \( x - 1 \):
Using synthetic division:
1 | 1 -6 11 -6
| 1 -5 6
-----------------------
1 -5 6 0
The result gives \( x^2 - 5x + 6 \), which we already factored as \( (x - 2)(x - 3) \).
So we have found the roots: \( x = 1, 2, \) and \( 3 \).
4. Graphical Method
For polynomials of any degree, graphing the function can give a good visual idea of where the roots are located. Roots are where the graph intersects the x-axis.
5. Numerical Methods
For complex polynomials or those that do not easily lend themselves to analytical methods, numerical methods (like Newton's method) can find approximate roots.
Conclusion
The method you choose depends on the polynomial's degree and structure. For degree 2 (quadratic), use factoring or the quadratic formula. For degree 3 or higher, use synthetic division, the Rational Root Theorem, or numerical methods as needed.