Polynomial
      Terms and the degree of a polynomial
      Addition and subtraction of polynomials
      Multiplication of polynomials
      Division of polynomials
Polynomial
Terms and the degree of a polynomial
The polynomial is mathematical expression consisting of a sum of terms each of which is a product of a constant and one or more variables raised to a non-negative integral power. If there is only a single variable x, the general form is given by
                      anxn + an-1xn-1 + an-2xn-2 + . . . + a2x2 + a1x + a0,
where the ai are real numbers called coefficients and a0 is the constant term.
The degree of a polynomial is the highest power or the sum of powers in any term of a given polynomial or algebraic expression.
Addition and subtraction of polynomials
We add or subtract polynomials by combining their like terms. The like terms are terms that have the same variables raised to the same exponents.
Example:   a( - 5x3 + 2x2 - x + 4) + ( - 4x2 + 3x - 7) = - 5x3 + ( 2 - 4) x2  + ( -1 + 3) x + 4 - 7 =
                                                                                    = - 5x3 - 2x2  + 2x - 3
                  b( x4 - 3x3 + 5x - 1) - ( - 2x4 + x3 - 3x2 + 4) = x4 - 3x3 + 5x - 1 + 2x4 - x3 + 3x2 - 4 =
                       = ( 1 + 2) x4  + ( -3 - 1) x3 + 3x2 + 5x - 5 = 3x - 4x3 + 3x2 + 5x - 5
Multiplication of polynomials
When multiplying two polynomials together, multiply every term of one polynomial by every term of the other polynomial using distributive property.
Example:   ( - 2x3 + 5x2 - x + 1) ( 3x - 2) =
        = 3x ( - 2x3) + 3x 5x2 + 3x   (-x) + 3x 1 + (- 2) ( - 2x3) + (- 2) 5x2 + (- 2) ( -x) + (- 2) 1 =
        = - 6x4 + 15x3 - 3x2 + 3x + 4x3 - 10x2 + 2x - 2 = - 6x4 + 19x3 - 13x2 + 5x - 2
Division of polynomials
Divide the highest degree term of dividend by the highest degree term of the divisor to get the first term of the quotient.
Take the first term of the quotient and multiply it by every term of divisor. Write this result below the dividend, making sure you line up all the terms with the terms of the dividend that has the same degree. 
Subtract the result from the dividend, i.e., reverse all the signs of the terms of the result and add like terms.
Repeat the process of long division until the degree of the new obtained dividend is less than the degree of the divisor.
Examples:
 
 
 
 
 
 
Note, since each second line should be subtracted, the sign of each term is reversed.
                       b 3x4 + x2 + 5) ( x2 - x - 1) = 3x2 + 3 x + 7     
 
 
or
 
   
like     17   5 = 3 + 2/5 
  -15 
      2
Beginning Algebra Contents C
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