## Roots of Polynomials

This example shows several different methods to calculate the roots of a polynomial.

### Numeric Roots

The `roots`

function calculates
the roots of a single-variable polynomial represented by a vector
of coefficients.

For example, create a vector to represent the polynomial $${x}^{2}-x-6$$, then calculate the roots.

p = [1 -1 -6]; r = roots(p)

r = 3 -2

By convention, MATLAB^{®} returns the roots in a column vector.

The `poly`

function converts
the roots back to polynomial coefficients. When operating on vectors, `poly`

and `roots`

are
inverse functions, such that `poly(roots(p))`

returns `p`

(up
to roundoff error, ordering, and scaling).

p2 = poly(r)

p2 = 1 -1 -6

When operating on a matrix, the `poly`

function
computes the characteristic polynomial of the matrix. The roots of
the characteristic polynomial are the eigenvalues of the matrix. Therefore, `roots(poly(A))`

and `eig(A)`

return
the same answer (up to roundoff error, ordering, and scaling).

### Roots Using Substitution

You can solve polynomial equations involving trigonometric functions by simplifying the equation using a substitution. The resulting polynomial of one variable no longer contains any trigonometric functions.

For example, find the values of $$\theta $$ that solve the equation

$$3{\mathrm{cos}}^{2}(\theta )-\mathrm{sin}(\theta )+3=0.$$

Use the fact that $${\mathrm{cos}}^{2}(\theta )=1-{\mathrm{sin}}^{2}(\theta )$$ to express the equation entirely in terms of sine functions:

$$-3{\mathrm{sin}}^{2}(\theta )-\mathrm{sin}(\theta )+6=0.$$

Use the substitution $$x=\mathrm{sin}(\theta )$$ to express the equation as a simple polynomial equation:

$$-3{x}^{2}-x+6=0.$$

Create a vector to represent the polynomial.

p = [-3 -1 6];

Find the roots of the polynomial.

r = roots(p)

`r = `*2×1*
-1.5907
1.2573

To undo the substitution, use $$\theta ={\mathrm{sin}}^{-1}(x)$$. The `asin`

function calculates the inverse sine.

theta = asin(r)

`theta = `*2×1 complex*
-1.5708 + 1.0395i
1.5708 - 0.7028i

Verify that the elements in `theta`

are the values of $$\theta $$ that solve the original equation (within roundoff error).

f = @(Z) 3*cos(Z).^2 - sin(Z) + 3; f(theta)

ans =2×1 complex10^{-14}× -0.0888 + 0.0647i 0.2665 + 0.0399i

### Roots in a Specific Interval

Use the `fzero`

function to find the roots of a polynomial in a specific interval. Among other uses, this method is suitable if you plot the polynomial and want to know the value of a particular root.

For example, create a function handle to represent the polynomial $$3{x}^{7}+4{x}^{6}+2{x}^{5}+4{x}^{4}+{x}^{3}+5{x}^{2}$$.

p = @(x) 3*x.^7 + 4*x.^6 + 2*x.^5 + 4*x.^4 + x.^3 + 5*x.^2;

Plot the function over the interval $$[-2,1]$$.

x = -2:0.1:1; plot(x,p(x)) ylim([-100 50]) grid on hold on

From the plot, the polynomial has a trivial root at `0`

and another near `-1.5`

. Use `fzero`

to calculate and plot the root that is near `-1.5`

.

Z = fzero(p, -1.5)

Z = -1.6056

`plot(Z,p(Z),'r*')`

### Symbolic Roots

If you have Symbolic Math Toolbox™, then there are additional
options for evaluating polynomials symbolically. One way is to use
the `solve`

(Symbolic Math Toolbox) function.

```
syms x
s = solve(x^2-x-6)
```

s = -2 3

Another way is to use the `factor`

(Symbolic Math Toolbox) function
to factor the polynomial terms.

F = factor(x^2-x-6)

F = [ x + 2, x - 3]

See Solve Algebraic Equations (Symbolic Math Toolbox) for more information.