a polynomial class for java implementing the basic operations +, -, /, *, compose and integrate.

Polynomial.java

// see http://introcs.cs.princeton.edu/java/92symbolic/Polynomial.java.html 

public class Polynomial{
    private int[] coef;  // coefficients
    private int deg;     // degree of polynomial (0 for the zero polynomial)


    // a * x^b
    public Polynomial( int a, int b ){
        coef = new int[ b + 1 ];
        coef[ b ] = a;
        deg = degree();
    }


    public Polynomial( Polynomial p ){
        coef = new int[ p.coef.length ];
        for( int i = 0; i < p.coef.length; i++ ){
            coef[ i ] = p.coef[ i ];
        }//end for
        deg = p.degree();
    }


    // return the degree of this polynomial (0 for the zero polynomial)
    public int degree(){
        int d = 0;
        for( int i = 0; i < coef.length; i++ )
            if( coef[ i ] != 0 ) d = i;
        return d;
    }


    // return c = a + b
    public Polynomial plus( Polynomial b ){
        Polynomial a = this;
        Polynomial c = new Polynomial( 0, Math.max( a.deg, b.deg ) );
        for( int i = 0; i <= a.deg; i++ ) c.coef[ i ] += a.coef[ i ];
        for( int i = 0; i <= b.deg; i++ ) c.coef[ i ] += b.coef[ i ];
        c.deg = c.degree();
        return c;
    }


    // return (a - b)
    public Polynomial minus( Polynomial b ){
        Polynomial a = this;
        Polynomial c = new Polynomial( 0, Math.max( a.deg, b.deg ) );
        for( int i = 0; i <= a.deg; i++ ) c.coef[ i ] += a.coef[ i ];
        for( int i = 0; i <= b.deg; i++ ) c.coef[ i ] -= b.coef[ i ];
        c.deg = c.degree();
        return c;
    }


    // return (a * b)
    public Polynomial times( Polynomial b ){
        Polynomial a = this;
        Polynomial c = new Polynomial( 0, a.deg + b.deg );
        for( int i = 0; i <= a.deg; i++ )
            for( int j = 0; j <= b.deg; j++ )
                c.coef[ i + j ] += ( a.coef[ i ] * b.coef[ j ] );
        c.deg = c.degree();
        return c;
    }


    // get the coefficient for the highest degree
    public int coeff(){return coeff( degree() ); }


    // get the coefficient for degree d
    public int coeff( int degree ){
        if( degree > this.degree() ) throw new RuntimeException( "bad degree" );
        return coef[ degree ];
    }


    /*
     Implement the division as described in wikipedia
        function n / d:
          require d ≠ 0
          q ← 0
          r ← n       # At each step n = d × q + r
          while r ≠ 0 AND degree(r) ≥ degree(d):
          t ← lead(r)/lead(d)     # Divide the leading terms
          q ← q + t
          r ← r − t * d
          return (q, r)
     */
    public Polynomial[] divides( Polynomial b ){
        Polynomial q = new Polynomial( 0, 0 );
        Polynomial r = new Polynomial( this );
        while( !r.isZero() && r.degree() >= b.degree() ){
            int coef = r.coeff() / b.coeff();
            int deg = r.degree() - b.degree();
            Polynomial t = new Polynomial( coef, deg );
            q = q.plus( t );
            r = r.minus( t.times( b ) );
        }//end while

        System.out.printf( "(%s) / (%s): %s, %s", this, b, q, r );
        return new Polynomial[]{ q, r };
    }


    // return a(b(x))  - compute using Horner's method
    public Polynomial compose( Polynomial b ){
        Polynomial a = this;
        Polynomial c = new Polynomial( 0, 0 );
        for( int i = a.deg; i >= 0; i-- ){
            Polynomial term = new Polynomial( a.coef[ i ], 0 );
            c = term.plus( b.times( c ) );
        }
        return c;
    }


    // do a and b represent the same polynomial?
    public boolean eq( Polynomial b ){
        Polynomial a = this;
        if( a.deg != b.deg ) return false;
        for( int i = a.deg; i >= 0; i-- )
            if( a.coef[ i ] != b.coef[ i ] ) return false;
        return true;
    }


    // test wether or not this polynomial is zero
    public boolean isZero(){
        for( int i : coef ){
            if( i != 0 ) return false;
        }//end for
        return true;
    }


    // use Horner's method to compute and return the polynomial evaluated at x
    public int evaluate( int x ){
        int p = 0;
        for( int i = deg; i >= 0; i-- )
            p = coef[ i ] + ( x * p );
        return p;
    }


    // differentiate this polynomial and return it
    public Polynomial differentiate(){
        if( deg == 0 ) return new Polynomial( 0, 0 );
        Polynomial deriv = new Polynomial( 0, deg - 1 );
        deriv.deg = deg - 1;
        for( int i = 0; i < deg; i++ )
            deriv.coef[ i ] = ( i + 1 ) * coef[ i + 1 ];
        return deriv;
    }


    // convert to string representation
    public String toString(){
        if( deg == 0 ) return "" + coef[ 0 ];
        if( deg == 1 ) return coef[ 1 ] + "x + " + coef[ 0 ];
        String s = coef[ deg ] + "x^" + deg;
        for( int i = deg - 1; i >= 0; i-- ){
            if( coef[ i ] == 0 ){
                continue;
            }else if( coef[ i ] > 0 ){
                s = s + " + " + ( coef[ i ] );
            }else if( coef[ i ] < 0 ) s = s + " - " + ( -coef[ i ] );
            if( i == 1 ){
                s = s + "x";
            }else if( i > 1 ) s = s + "x^" + i;
        }
        return s;
    }


    // test client
    public static void main( String[] args ){
        Polynomial zero = new Polynomial( 0, 0 );

        Polynomial p1 = new Polynomial( 1, 3 );
        Polynomial p2 = new Polynomial( 2, 2 );
        Polynomial p3 = new Polynomial( 4, 0 );
        Polynomial p4 = new Polynomial( 0, 1 );
        Polynomial p = p1.plus( p2 ).plus( p3 ).plus( p4 );   // 4x^3 + 3x^2 + 1

        Polynomial q1 = new Polynomial( 1, 1 );
        Polynomial q2 = new Polynomial( 3, 0 );
        Polynomial q = q1.plus( q2 );                     // 3x^2 + 5


        Polynomial r = p.plus( q );
        Polynomial s = p.times( q );
        Polynomial t = p.compose( q );

        System.out.println( "zero(x) =     " + zero );
        System.out.println( "p(x) =        " + p );
        System.out.println( "q(x) =        " + q );
        System.out.println( "p(x) + q(x) = " + r );
        System.out.println( "p(x) * q(x) = " + s );
        System.out.println( "p(q(x))     = " + t );
        System.out.println( "0 - p(x)    = " + zero.minus( p ) );
        System.out.println( "p(3)        = " + p.evaluate( 3 ) );
        System.out.println( "p'(x)       = " + p.differentiate() );
        System.out.println( "p''(x)      = " + p.differentiate().differentiate() );

        p.divides( q );
    }

}