How to Divide Polynomials

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Polynomials can be divided the same as numeric constants, either by factoring or by long division. The method you use depends upon how complex the polynomial dividend and divisor are.

Part 1
Part 1 of 3:

Determining Which Approach to Use

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  1. How complicated the divisor (the polynomial you’re dividing by) is compared to the dividend (the polynomial you’re dividing into) determines which approach is best.
    • If the divisor is a monomial (single-term polynomial), either a variable with a coefficient, or a constant (a number without a variable following it), you can probably factor the dividend and cancel out one of the resulting factors and the divisor. See “Factoring the Dividend” for instructions and examples.
    • If the divisor is a binomial (two-term polynomial), you may be able to factor the dividend and cancel out one of the resulting factors and the divisor.
    • If the divisor is a trinomial (three-term polynomial), you may be able to factor both the dividend and the divisor, cancel out the common factor, and then either factor out the dividend further or use long division.
    • If the divisor is a polynomial with more than three factors, you will probably have to use long division. [1] See “Using Long Polynomial Division” for instructions and examples.
  2. If looking at the divisor polynomial of the equation doesn’t tell you whether you should try to factor the dividend, look at the dividend itself. [2]
    • If the dividend has three terms or fewer, you can probably factor it and cancel out the divisor. [3]
    • If the dividend has more than three terms, you will probably have to divide the divisor into it by using long division.
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Part 2
Part 2 of 3:

Factoring the Dividend [4]

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  1. If this is the case, you can factor it out and probably cancel out the divisor. [5]
    • If you’re dividing the binomial 3x – 9 by 3, you can factor 3 out of both terms of the binomial, making it 3(x – 3). You can then cancel out the divisor of 3, leaving a quotient of x – 3.
    • If you’re dividing the binomial 24x 3 - 18x 2 by 6x, you can factor 6x out of both terms of the binomial, making it 6x(4x 2 - 3). You can the cancel out the divisor of 6x, leaving a quotient of 4x 2 - 3.
  2. Certain polynomials display terms that tell you they can be factored. If one of those factors matches the divisor, you can cancel it out, leaving the remaining factor as the quotient. Here are some patterns to look for:
    • Difference of perfect squares. This is a binomial of the form ‘’a 2 x 2 - b 2 ’’, where the values of ‘’a 2 ’’ and ‘’b 2 ’’ are perfect squares. This binomial factors into two binomials (ax + b)(ax – b), where a and b are the square roots of the coefficient and constant of the earlier binomial.
    • Perfect square trinomial. This trinomial is in the form a 2 x 2 + 2abx + b 2 . It factors to (ax + b)(ax + b), which may also be written (ax + b) 2 . If the sign in front of the second term is a minus sign, the binomial factors will be in the form (ax – b)(ax – b).
    • Sum or difference of cubes. This is a binomial of the form a 3 x 3 + b 3 or a 3 x 3 - b 3 , where the values of ‘’a 3 ’’ and ‘’b 3 ’’ are perfect cubes. This binomial factors into a binomial and a trinomial. A sum of cubes factors down to (ax + b)(a 2 x 2 - abx + b 2 ). A difference of cubes factors down to (ax - b)(a 2 x 2 + abx + b 2 ).
  3. If you don’t see a discernible pattern in the dividend to tell you how to factor it, you can try several possible factoring combinations. You can do this by looking first at the constant and finding several factors for it, then at the coefficient of the middle term.
    • For example, if the dividend is x 2 - 3x – 10, you would look at the factors of 10 and use the 3 to help determine which factor pair is correct.
    • The number 10 can be broken into factors of 1 and 10 or 2 and 5. Because the sign in front of 10 is negative, one of the factor binomials has to have a negative number in front of its constant.
    • The number 3 is the difference between 2 and 5, so these must be the constants of the factor binomials. Because the sign in front of the 3 is negative, the binomial with the 5 must be the one with the negative number. The binomial factors are thus (x – 5)(x + 2). If the divisor is one of these two factors, that factor can be canceled out, and the remaining factor is the quotient.
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Part 3
Part 3 of 3:

Using Long Polynomial Division [6]

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  1. You write out the long division of polynomials the same as you do for dividing numbers. The dividend goes under the long division bar, while the divisor goes to the left. [7]
    • If you’re dividing x 2 + 11 x + 10 by x +1, x 2 + 11 x + 10 goes under the bar, while x + 1 goes to the left.
  2. The result of this division goes on top of the division bar. [8]
    • For our example, dividing x 2 , the first term of the dividend, by x, the first term of the divisor yields x. You would write an x on the top of the division bar, over the x 2 .
  3. Write the result of the multiplication under the leftmost terms of the dividend. [9]
    • Continuing with our example, multiplying x + 1 by x produces x 2 + x. You would write this under the first two terms of the dividend.
  4. To do so, first reverse the signs of the product of the multiplication. After subtracting, bring down the remaining terms of the dividend. [10]
    • Reversing the signs of x 2 + x gives - x 2 - x. Subtracting this from the first two terms of the dividend leaves 10x. After bringing down the remaining term of the dividend, you have 10x + 10 as the interim quotient to continue the division process with.
  5. You’ll again divide the first term of the divisor into that of the interim quotient, write that result on top of the division bar after the first term of the quotient, multiply the result by the divisor, and then calculate what to subtract from the interim quotient. [11]
    • Because x goes into 10x 10 times, you would write “+ 10” after the x in the quotient position on the division bar.
    • Multiplying x +1 by 10 gives 10x + 10. You write this under the interim quotient and reverse the signs for the subtraction, making -10x – 10.
    • When you perform the subtraction, you have a remainder of 0. Thus, dividing x 2 + 11 x + 10 by x +1 produces a quotient of x + 10. (You could have gotten the same result by factoring, but this example was chosen to keep the division fairly simple.)
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  • Question
    What if there is no constant present in the polynomial?
    Orangejews
    Community Answer
    You can always add + 0 to the polynomial if you find having that placeholder helpful. Otherwise, the same steps all work, just remember 0 times anything is 0.
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  • Question
    The polynomial 2x3+x2-3x+p has a reminder 20 when divided by (x-2), what is the value of constant p?
    Community Answer
    Use the Remainder Theorem to turn this into an easier problem: Call that polynomial f(x), then the Remainder Theorem implies f(2) = 20. f(2) = 2*(8) + 4 - 3*(2) + p = 14 + p. If 14 + p = 20, then p = 6 and you don't actually have to work through an ugly polynomial division.
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  • Question
    Given the polynomial Q(x) =x^3 +y^3 where y is real, how so I find P(-y) then factor x^3+y^3 as a product of linear and quadratic factor?
    Community Answer
    Factor Q using the difference of cubes factorization trick. But isn't Q is a sum of cubes, not a difference so the trick doesn't apply here? No problem, because we can consider -y instead of +y and then it is the difference of cubes! Q(x) = x^3 + y^3 = x^3 - (-y)^3 = [x-(-y)]*[x^2 + x(-y) + (-y)^2] = (x+y)(x^2 - xy + y^2).
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      Tips

      • If, when performing long division on a polynomial, you have a non-zero remainder, you can make that remainder part of the quotient by writing it as a fraction using the remainder as the numerator and the divisor as the denominator. [12] If, in our long division example, the dividend had been x 2 + 11 x + 12 instead of x 2 + 11 x + 10, dividing by x +1 would have left a remainder of 2. The complete quotient would thus be written as:
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      • Be aware that some algebra books format long polynomial division with the quotient and dividend right-justified, or with the terms presented such that the terms of the same degree within both polynomials are aligned with each other. You will probably find it easier, however, when doing the division by hand, to left-justify the quotient and dividend as described in the steps above.
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      • If your dividend has a gap in the degrees of its terms, such as 3x 3 +9x 2 +18, you can insert the missing term with a coefficient of 0, in this case, 0x to make it easier to position the other terms during the division. Doing this doesn’t change the dividend’s value.
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      Warnings

      • When writing the quotient of a polynomial division that includes a fractional term, always use a plus sign between the whole number (or whole variable) term and the fractional term.
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      • Keep your columns aligned when doing long polynomial division to avoid subtracting the wrong terms from each other.
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      References

      1. ↑ https://courses.lumenlearning.com/wmopen-collegealgebra/chapter/introduction-dividing-polynomials/
      2. ↑ https://www.cuemath.com/algebra/dividing-polynomials/
      3. ↑ https://www.mathsisfun.com/algebra/polynomials-remainder-factor.html
      4. ↑ http://www.purplemath.com/modules/polydiv.htm
      5. ↑ https://www.cuemath.com/algebra/dividing-polynomials/
      6. ↑ http://www.purplemath.com/modules/polydiv2.htm
      7. ↑ https://www.mathsisfun.com/algebra/polynomials-division-long.html
      8. ↑ https://www.cuemath.com/algebra/long-division-of-polynomials/
      9. ↑ https://www.mathsisfun.com/algebra/polynomials-division-long.html
      1. ↑ https://www.mathsisfun.com/algebra/polynomials-division-long.html
      2. ↑ https://www.cuemath.com/algebra/long-division-of-polynomials/
      3. ↑ http://www.purplemath.com/modules/polydiv3.htm

      About This Article

      Article Summary X

      To divide polynomials, start by writing out the long division of your polynomial the same way you would for numbers. For example, put the dividend under the long division bar and the diviser to the left. Then, divide the first term of the divisor into the first term of the dividend, and multiply the X in the quotient by the divisor. Finally, subtract from the dividend before repeating the previous 3 steps on the interim quotient. For tips on how to determine whether to use long division or factoring for your polynomial, read on!

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