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Subtracting Polynomials (page 2 of 2)

Sections: Adding polynomials, Subtracting polynomials

Subtracting polynomials is quite similar to adding polynomials, but you have that pesky minus sign to deal with. Here are some examples, done both horizontally and vertically:

  • Simplify (x3 + 3x2 + 5x 4) (3x3 8x2 5x + 6)

    The first thing I have to do is take that negative through the parentheses. Some students find it helpful to put a "1" in front of the parentheses, to help them keep track of the minus sign:


      (x3 + 3x2 + 5x 4) (3x3 8x2 5x + 6)
        =  (x3 + 3x2 + 5x 4)
      1(3x3 8x2 5x + 6)
        =  (x3 + 3x2 + 5x 4) 1(3x3) 1 (8x2) 1(5x) 1(6)
        =  x3 + 3x2 + 5x 4 3x3 + 8x2 + 5x 6

        =  x3 3x3 + 3x2 + 8x2 + 5x + 5x 4 6

      2x3 + 11x2 + 10x 10

    Vertically:   Copyright Elizabeth Stapel 2000-2011 All Rights Reserved


In the horizontal case, you may have noticed that running the negative through the parentheses changed the sign on each term inside the parentheses. The shortcut here is to not bother writing in the subtaction sign or the parentheses; instead, you just change all the signs in the second row.

    I'll change all the signs in the second row (shown in red below), and add down:

      change signs

    Either way, I get the answer:  2x3 + 11x2 + 10x 10

  • Simplify (6x3 2x2 + 8x) (4x3 11x + 10)


      (6x3 2x2 + 8x) (4x3 11x + 10)
        =  (6x3 2x2 + 8x)
      1(4x3 11x + 10)
        =  (6x3 2x2 + 8x) 1(4x3) 1(11x) 1(10)
        =  6x3 2x2 + 8x 4x3 + 11x 10

        =  6x3 4x3 2x2 + 8x + 11x 10

      2x3 2x2 + 19x 10

    Vertically: I'll write out the polynomials, leaving gaps as necessary:

      Write out the polynomials.

    Then I'll change the signs in the second line, and add:

      Change the signs on the second row, and add down.

    Either way, I get the answer: 2x3 2x2 + 19x 10

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Cite this article as:

Stapel, Elizabeth. "Subtracting Polynomials." Purplemath. Available from Accessed


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