Polynomial class makes possible to do simple operations like additions, subtractions and multiplications on polynomials.

Polynomial.php

<?php
/**
 * Polynomial class makes possible to do simple operations like additions, subtractions and
 * multiplications on polynomials.
 *
 * There are two ways to initiate a Polynomial object: from string or from array.
 *
 * @package fintara
 * @author  Tsvetan Ovedenski <[email protected]>
 */

namespace fintara;

class Polynomial {

    private $_varLetter;
    private $_coefficients = array();

    /**
     * @param char $varLetter
     */
    public function __construct($varLetter = "x") {
        if(strlen($varLetter) !== 1)
            throw new \Exception("The variable letter should be only 1 character.");

        $this->_varLetter = strtolower($varLetter);
    }

    /**
     * Sets a polynomial from a string.
     * Example format: x^3 + 2x - 3
     * Whitespaces don't matter.
     *
     * @param  string $polynomial
     * @return Polynomial
     */
    public function setFromString($polynomial) {
        $polynomial = str_replace(" ", "", $polynomial);
        $this->setCoefficientsFromString($polynomial);
        return $this;
    }

    /**
     * Sets a polynomial from an array.
     *
     * @param  array $polynomial
     * @return Polynomial
     */
    public function setFromArray(array $polynomial) {
        krsort($polynomial);
        $this->_coefficients = $polynomial;
        return $this;
    }

    /**
     * Gets the coefficients from a string and sets them in the array.
     *
     * @param  string $input
     * @return void
     */
    private function setCoefficientsFromString($input) {
        $input = str_replace("-", "+-", $input);
        $parts = explode("+", $input);

        foreach ($parts as $part) {
            if (preg_match("/[\d+]{$this->_varLetter}\^[\d+]/", $part)) {
                list($coeff, $power) = explode("^", $part);
                $this->_coefficients[(int) $power] = (int) str_replace($this->_varLetter, "", $coeff);
            } else if (preg_match("/-{$this->_varLetter}\^[\d+]/", $part)) {
                list($coeff, $power) = explode("^", $part);
                $this->_coefficients[(int) $power] = -1;
            } else if (preg_match("/{$this->_varLetter}\^[\d+]/", $part)) {
                list($coeff, $power) = explode("^", $part);
                $this->_coefficients[(int) $power] = 1;
            }else if (preg_match("/[\d+]{$this->_varLetter}/", $part)) {
                $this->_coefficients[1] = (int) str_replace($this->_varLetter, "", $part);
            } else if($part === "-{$this->_varLetter}") {
                $this->_coefficients[1] = -1;
            } else if($part === $this->_varLetter) {
                $this->_coefficients[1] = 1;
            } else {
                $this->_coefficients[0] = (int) $part;
            }
        }

        for($i = 0; $i <= $this->getMaxDegree(); $i++) {
            if(!isset($this->_coefficients[$i])) {
                $this->_coefficients[$i] = 0;
            }
        }
    }

    /**
     * @return array
     */
    public function getCoefficients()
    {
        return $this->_coefficients;
    }

    /**
     * Returns the max degree of the polynomial.
     * @return integer
     */
    public function getMaxDegree() {
        ksort($this->_coefficients);
        end($this->_coefficients);
        $maxDegree = key($this->_coefficients);
        reset($this->_coefficients);
        return $maxDegree;
    }

    /**
     * Returns the polynomial in a readable output.
     * Example output: x^3+2x-1
     *
     * @param  char $varLetter
     * @return string
     */
    public function getBeautifulOutput($varLetter = null) {
        if(!$varLetter)
            $varLetter = $this->_varLetter;

        krsort($this->_coefficients);

        $output = "";

        $i = 0;

        foreach($this->_coefficients as $power => $value) {
            if($value !== 0) {

                if(abs($value) !== 1 || $power === 0)
                    $output .= $value;
                else if($value === -1)
                    $output .= "-";

                if($value > 0 && $i !== 0)
                    $output .= "+";

                if($power > 0) {
                    $output .= $varLetter;

                    if($power > 1)
                        $output .= "^{$power}";
                }
            }

            $i++;
        }

        return $output;

    }

    /**
     * Add a polynomial.
     *
     * @param  Polynomial $polynomial
     * @return Polynomial
     */
    public function add(Polynomial $polynomial) {
        $guestCoefficients = $polynomial->getCoefficients();

        $maxDegree = max($polynomial->getMaxDegree(), $this->getMaxDegree());

        for($i = 0; $i <= $maxDegree; $i++) {
            if(!isset($guestCoefficients[$i])) {
                $guestCoefficients[$i] = $this->_coefficients[$i];
            } else if(isset($this->_coefficients[$i])) {
                $guestCoefficients[$i] += $this->_coefficients[$i];
            }
        }

        return (new Polynomial())->setFromArray($guestCoefficients);
    }

    /**
     * Subtract a polynomial.
     *
     * @param  Polynomial $polynomial
     * @return Polynomial
     */
    public function subtract(Polynomial $polynomial) {
        $guestCoefficients = $polynomial->getCoefficients();
        $thisCoefficients  = $this->_coefficients;

        $maxDegree = max($polynomial->getMaxDegree(), $this->getMaxDegree());

        for($i = 0; $i <= $maxDegree; $i++) {
            if(!isset($guestCoefficients[$i])) {
                $thisCoefficients[$i] = 0 - $guestCoefficients[$i];
            } else if(isset($guestCoefficients[$i])) {
                $thisCoefficients[$i] -= $guestCoefficients[$i];
            }
        }

        krsort($thisCoefficients);
        return (new Polynomial())->setFromArray($thisCoefficients);
    }

    /**
     * Multiply a polynomial.
     *
     * @param  Polynomial $polynomial
     * @return Polynomial
     */
    public function multiply(Polynomial $polynomial) {
        $guestCoefficients = $polynomial->getCoefficients();
        $newCoefficients   = array_fill(0, $this->getMaxDegree() + $polynomial->getMaxDegree() + 1, 0);

        foreach($this->_coefficients as $tPower => $tValue) {
            foreach($guestCoefficients as $gPower => $gValue) {
                $tmpNewPower = $tPower + $gPower;

                $newCoefficients[$tmpNewPower] += $tValue * $gValue;
            }
        }

        return (new Polynomial())->setFromArray($newCoefficients);
    }
}

$polynomA = (new Polynomial('x'))->setFromString('-x');
$polynomB = (new Polynomial('x'))->setFromString('x-1x^3');

echo $polynomA->getBeautifulOutput(); echo "\n";
echo $polynomB->getBeautifulOutput(); echo "\n\n";

echo $polynomA->multiply($polynomB)->getBeautifulOutput('y');