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8.10: Dividing Polynomials

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    60052
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    Dividing A Polynomial By A Monomial

    The following examples illustrate how to divide a polynomial by a monomial. The division process is quite simple and is based on the addition of rational expressions.

    \(\dfrac{a}{c} + \dfrac{b}{c} = \dfrac{a+b}{c}\)

    Turning this equation around we get

    \(\dfrac{a+b}{c} = \dfrac{a}{c} + \dfrac{b}{c}\)

    Now we simply divide \(c\) into \(a\), and \(c\) into \(b\). This should suggest a rule.

    Dividing a Polynomial By a Monomial

    To divide a polynomial by a monomial, divide every term of the polynomial by the monomial.

    Sample Set A

    Example \(\PageIndex{1}\)

    \(\dfrac{3x^2 + x - 11}{x}\). Divide every term of \(3x^2 + x - 11\) by \(x\).

    \(\dfrac{3x^2}{x} + \dfrac{x}{x} - \dfrac{11}{x} = 3x + 1 - \dfrac{11}{x}\)

    Example \(\PageIndex{2}\)

    \(\dfrac{8x^3 + 4a^2 - 16a + 9}{2a^2}. Divide every term of \(8a^3 + 4a^2 - 16a + 9\) by \(2a^2\).

    Example \(\PageIndex{3}\)

    \(\dfrac{4b^6 - 9b^4 - 2b + 5}{-4b^2}\). Divide every term of \(4b^6 - 9b^4 - 2b + 5\) by \(-4b^2\).

    \(\dfrac{4b^6}{-4b^2} - \dfrac{9b^4}{-4b^2} - \dfrac{2b}{-4b^2} + \dfrac{5}{-4b^2} = -b^4 + \dfrac{9}{4}b^2 + \dfrac{1}{2b} - \dfrac{5}{4b^2}\)

    Practice Set A

    Perform the following divisions.

    Practice Problem \(\PageIndex{1}\)

    \(\dfrac{2x^2 + x - 1}{x}\)

    Answer

    \(2x + 1 - \dfrac{1}{x}\)

    Practice Problem \(\PageIndex{2}\)

    \(\dfrac{3x^3 + 4x^2 + 10x - 4}{x^2}\)

    Answer

    \(3x + 4 + \dfrac{10}{x} - \dfrac{4}{x^2}\)

    Practice Problem \(\PageIndex{3}\)

    \(\dfrac{a^2b + 3ab^2 + 2b}{ab}\)

    Answer

    \(a + 3b + \dfrac{2}{a}\)

    Practice Problem \(\PageIndex{4}\)

    \(\dfrac{14x^2y^2 - 7xy}{7xy}\)

    Answer

    \(2xy−1\)

    Practice Problem \(\PageIndex{5}\)

    \(\dfrac{10m^3n^2 + 15m^2n^3 - 20mn}{-5m}\)

    Answer

    \(-2m^2n^2 - 3mn^3 + 4n\)

    The Process Of Division

    In Section 8.3 we studied the method of reducing rational expressions. For example, we observed how to reduce an expression such as

    \(\dfrac{x^2 - 2x - 8}{x^2 - 3x - 4}\)

    Our method was to factor both the numerator and denominator, then divide out common factors.

    \(\dfrac{(x-4)(x+2)}{(x-4)(x+1)}\)

    \(\dfrac{\cancel{(x-4)}(x+2)}{\cancel{(x-4)}(x+1)}\)

    \(\dfrac{x+2}{x+1}\)

    When the numerator and denominator have no factors in common, the division may still occur, but the process is a little more involved than merely factoring. The method of dividing one polynomial by another is much the same as that of dividing one number by another. First, we’ll review the steps in dividing numbers.

    \(\dfrac{35}{8}\). We are to divide 35 by 8.

    Long division showing eight dividing thirty five. This division is not performed completely. We try 4, since 32 divided by 8 is 4.

    Long division showing eight dividing thirty five, with four at quotient's place. This division is not performed completely. Multiply 4 and 8

    Long division showing eight dividing thirty five, with four at quotient's place. Thirty two is written under thirty five. This division is not performed completely Subtract 32 from 35

    Long division showing eight dividing thirty five, with four at quotient's place. Thirty two is written under thirty five and three is written as the subtraction of thirty five and thirty two.Since the remainder 3 is less than the divisor 8, we are done with the 32 division.

    \(4\dfrac{3}{8}\). The quotient is expressed as a mixed number.

    The process was to divide, multiply, and subtract.

    Review Of Subtraction Of Polynomials

    A very important step in the process of dividing one polynomial by another is the subtraction of polynomials. Let’s review the process of subtraction by observing a few examples.

    1. Subtract \(x -2\) from \(x-5\); that is, find \((x-5) - (x-2)\).

    Since \(x-2\) is preceded by a minus sign, remove the parentheses, change the sign of each term, then add.

    \(\begin{array}{flushleft}
    x-5 && x-5\\
    -(x-2) && -x+2\\
    \text{_______} & = & \text{_______}\\
    &&-3
    \end{array}\)

    The result is \(-3\)

    2. Subtract \(x^3 + 3x^2\) from \(x^3 + 4x^2 + x - 1\).

    Since \(x^3 + 3x^2\) is preceded by a minus sign, remove the parentheses, change the sign of each term, then add.

    \(\begin{array}{flushleft}
    x^3 + 4x^2 + x - 1 && x^3 + 4x^2 + x - 1\\
    -(x^3 + 3x^2) && -x^3 - 3x^2\\
    \text{_______________} & = & \text{_______________}\\
    &&x^2 + x - 1
    \end{array}\)

    The result is \(x^2 + x - 1\)

    3. Subtract \(x^2 + 3x\) from \(x^2 + 1\)

    We can write \(x^2 + 1\) as \(x^2 + 0x + 1\).

    \(\begin{array}{flushleft}
    x^2 + 1 && x^2 + 0x + 1 && x^2 + 0x + 1\\
    -(x^2 + 3x) && -(x^2 + 3x) && -x^2 - 3x\\
    \text{____________} & = & \text{____________} & = & \text{____________}\\
    &&&& -3x + 1
    \end{array}\)

    Dividing A Polynomial By A Polynomial

    Now we’ll observe some examples of dividing one polynomial by another. The process is the same as the process used with whole numbers: divide, multiply, subtract, divide, multiply, subtract,....

    The division, multiplication, and subtraction take place one term at a time. The process is concluded when the polynomial remainder is of lesser degree than the polynomial divisor.

    Sample Set B

    Perform the division.

    Example \(\PageIndex{4}\)

    \(\dfrac{x-5}{x-2}\). We are to divide \(x-5\) by \(x-2\).

    Long division showing x minus two dividing x minus five with the comment 'Divide x into x' on the right side. This division is not performed completely. See the longdesc for a full description.

    \(1 - \dfrac{3}{x-2}\)

    Thus,

    \(\dfrac{x-5}{x-2} = 1 - \dfrac{3}{x-2}\)

    Example \(\PageIndex{5}\)

    \(\dfrac{x^3 + 4x^2 + x - 1}{x + 3}\). We are to divide \(x^3 + 4x^2 + x - 1\) by \(x + 3\).

    Long division showing x plus three dividing x cube plus four x square plus x minus one with the comment 'Divide x into x cube' on the right side. This division is not performed completely. See the longdesc for a full description

    \(x^2 + x - 2 + \dfrac{5}{x+3}\)

    Thus,

    \(\dfrac{x^3 + 4x^2 + x - 1}{x + 3} = x^2 + x - 2 + \dfrac{5}{x+3}\)

    Practice Set B

    Perform the following divisions.

    Practice Problem \(\PageIndex{6}\)

    \(\dfrac{x+6}{x-1}\)

    Answer

    \(1 + \dfrac{7}{x-1}\)

    Practice Problem \(\PageIndex{7}\)

    \(\dfrac{x^2 + 2x + 5}{x + 3}\)

    Answer

    \(x - 1 + \dfrac{8}{x+3}\)

    Practice Problem \(\PageIndex{8}\)

    \(\dfrac{x^3 + x^2 - x - 2}{x + 8}\)

    Answer

    \(x^2 - 7x + 55 - \dfrac{442}{x+8}\)

    Practice Problem \(\PageIndex{9}\)

    \(\dfrac{x^3 + x^2 - 3x + 1}{x^2 + 4x - 5}\)

    Answer

    \(x - 3 + \dfrac{14x - 14}{x^2 + 4x - 5} = x - 3 + \dfrac{14}{x+5}\)

    Sample Set C

    Example \(\PageIndex{6}\)

    Divide \(2x^3 - 4x + 1\) by \(x + 6\)

    \(\dfrac{2x^3 - 4x + 1}{x + 6}\) Notice that the \(x^2\) term in the numerator is missing. We can avoid any confusion by writing

    \(\dfrac{2x^3 + 0x^2 - 4x + 1}{x+6}\) Divide, multiply, and subtract.

    Steps of long division showing the quantity x plus six dividing the quantity two x cubed plus zero x squared minus four x minus plus one. See the longdesc for a full description

    \(\dfrac{2x^3 - 4x + 1}{x + 6} = 2x^3 - 12x + 68 - \dfrac{407}{x + 6}\)

    Practice Set C

    Perform the following divisions.

    Practice Problem \(\PageIndex{10}\)

    \(\dfrac{x^2 - 3}{x+2}\)

    Answer

    \(x - 2 + \dfrac{1}{x+2}\)

    Practice Problem \(\PageIndex{11}\)

    \(\dfrac{4x^2 - 1}{x-3}\)

    Answer

    \(4x + 12 + \dfrac{35}{x-3}\)

    Practice Problem \(\PageIndex{12}\)

    \(\dfrac{x^3 + 2x + 2}{x-2}\)

    Answer

    \(x^2 + 2x + 6 + \dfrac{14}{x-2}\)

    Practice Problem \(\PageIndex{13}\)

    \(\dfrac{6x^3 + 5x^2 - 1}{2x + 3}\)

    Answer

    \(3x^2 - 2x + 3 - \dfrac{10}{2x + 3}\)

    Exercises

    For the following problems, perform the divisions.

    Exercise \(\PageIndex{1}\)

    \(\dfrac{6a + 12}{2}\)

    Answer

    \(3a+6\)

    Exercise \(\PageIndex{2}\)

    \(\dfrac{12b - 6}{3}\)

    Exercise \(\PageIndex{3}\)

    \(\dfrac{8y - 4}{-4}\)

    Answer

    \(−2y+1\)

    Exercise \(\PageIndex{4}\)

    \(\dfrac{21a - 9}{-3}\)

    Exercise \(\PageIndex{5}\)

    \(\dfrac{3x^2 - 6x}{-3}\)

    Answer

    \(−x(x−2)\)

    Exercise \(\PageIndex{6}\)

    \(\dfrac{4y^2 - 2y}{2y}\)

    Exercise \(\PageIndex{7}\)

    \(\dfrac{9a^2 + 3a}{2a}\)

    Answer

    \(3a+1\)

    Exercise \(\PageIndex{8}\)

    \(\dfrac{20x^2 + 10x}{5x}\)

    Exercise \(\PageIndex{9}\)

    \(\dfrac{6x^3 + 2x^2 + 8x}{2x}\)

    Answer

    \(3x^2 + x + 4\)

    Exercise \(\PageIndex{10}\)

    \(\dfrac{26y^3 + 13y^2 + 39y}{13y}\)

    Exercise \(\PageIndex{11}\)

    \(\dfrac{a^2b^2 + 4a^2b + 6ab^2 - 10ab}{ab}\)

    Answer

    \(ab+4a+6b−10\)

    Exercise \(\PageIndex{12}\)

    \(\dfrac{7x^3y + 8x^2y^3 + 3xy^4 - 4xy}{xy}\)

    Exercise \(\PageIndex{13}\)

    \(\dfrac{5x^3y^3 - 15x^2y^2 + 20xy}{-5xy}\)

    Answer

    \(-x^2y^2 + 3xy - 4\)

    Exercise \(\PageIndex{14}\)

    \(\dfrac{4a^2b^3 - 8ab^4 + 12ab^2}{-2ab^2}\)

    Exercise \(\PageIndex{15}\)

    \(\dfrac{6a^2y^2 + 12a^2y + 18a^2}{24a^2}\)

    Answer

    \(\dfrac{1}{4}y^2 + \dfrac{1}{2}y + \dfrac{3}{4}\)

    Exercise \(\PageIndex{16}\)

    \(\dfrac{3c^3y^3 + 99c^3y^4 - 12c^3y^5}{3x^3y^3}\)

    Exercise \(\PageIndex{17}\)

    \(\dfrac{16ax^2 - 20ax^3 + 24ax^4}{6a^4}\)

    Answer

    \(\dfrac{8x^2 - 10x^3 + 12x^4}{3a^3}\) or \(\dfrac{12x^4 - 10x^3 + 8x^2}{3a^2}\)

    Exercise \(\PageIndex{18}\)

    \(\dfrac{21ay^3 - 18ay^2 - 15ay}{6ay^2}\)

    Exercise \(\PageIndex{19}\)

    \(\dfrac{-14b^2c^2 + 21b^3 - 28c^3}{-7a^2c^3}\)

    Answer

    \(\dfrac{2b^2 - 3b^3c + 4c}{a^2c}\)

    Exercise \(\PageIndex{20}\)

    \(\dfrac{-30a^2b^4 - 35a^2b^3 - 25a^2}{-5b^3}\)

    Exercise \(\PageIndex{21}\)

    \(\dfrac{x+6}{x-2}\)

    Answer

    \(1 + \dfrac{8}{x-2}\)

    Exercise \(\PageIndex{22}\)

    \(\dfrac{y + 7}{y + 1}\)

    Exercise \(\PageIndex{23}\)

    \(\dfrac{x^2 - x + 4}{x + 2}\)

    Answer

    \(x - 3 + \dfrac{10}{x+2}\)

    Exercise \(\PageIndex{24}\)

    \(\dfrac{x^2 + 2x - 1}{x + 1}\)

    Exercise \(\PageIndex{25}\)

    \(\dfrac{x^2 - x + 3}{x + 1}\)

    Answer

    \(x - 2 + \dfrac{5}{x + 1}\)

    Exercise \(\PageIndex{26}\)

    \(\dfrac{x^2 + 5x + 5}{x + 5}\)

    Exercise \(\PageIndex{27}\)

    \(\dfrac{x^2 - 2}{x + 1}\)

    Answer

    \(x - 1 - \dfrac{1}{x+1}\)

    Exercise \(\PageIndex{28}\)

    \(\dfrac{a^2 - 6}{a + 2}\)

    Exercise \(\PageIndex{29}\)

    \(\dfrac{y^2 + 4}{y + 2}\)

    Answer

    \(y - 2 + \dfrac{8}{y + 2}\)

    Exercise \(\PageIndex{30}\)

    \(\dfrac{x^2 + 36}{x + 6}\)

    Exercise \(\PageIndex{31}\)

    \(\dfrac{x^3 - 1}{x + 1}\)

    Answer

    \(x^2 - x + 1 - \dfrac{2}{x + 1}\)

    Exercise \(\PageIndex{32}\)

    \(\dfrac{a^3 - 8}{a + 2}\)

    Exercise \(\PageIndex{33}\)

    \(\dfrac{x^3 + 3x^2 + x - 2}{x-2}\)

    Answer

    \(x^2 + 5x + 11 + \dfrac{20}{x-2}\)

    Exercise \(\PageIndex{34}\)

    \(\dfrac{a^3 + 2a^2 - a + 1}{a - 3}\)

    Exercise \(\PageIndex{35}\)

    \(\dfrac{x^3 + 2x + 1}{x - 3}\)

    Exercise \(\PageIndex{36}\)

    \(\dfrac{y^3 + 2y^2 + 4}{y + 2}\)

    Answer

    \(y^2 + y - 2 + \dfrac{8}{y + 2}\)

    Exercise \(\PageIndex{37}\)

    \(\dfrac{y^3 + 5y^2 - 3}{y - 1}\)

    Exercise \(\PageIndex{38}\)

    \(\dfrac{x^3 + 3x^2}{x + 3}\)

    Answer

    \(x^2\)

    Exercise \(\PageIndex{39}\)

    \(\dfrac{a^2 + 2a}{a + 2}\)

    Exercise \(\PageIndex{40}\)

    \(\dfrac{x^2 - x - 6}{x^2 - 2x - 3}\)

    Answer

    \(1 + \dfrac{1}{x + 1}\)

    Exercise \(\PageIndex{41}\)

    \(\dfrac{a^2 + 5a + 4}{a^2 - a - 2}\)

    Exercise \(\PageIndex{42}\)

    \(\dfrac{2y^2 + 5y + 3}{y^2 - 3y - 4}\)

    Answer

    \(2 + \dfrac{11}{y-4}\)

    Exercise \(\PageIndex{43}\)

    \(\dfrac{3a^2 + 4a + 2}{3a + 4}\)

    Exercise \(\PageIndex{44}\)

    \(\dfrac{6x^2 + 8x - 1}{3x + 4}\)

    Answer

    \(2x - \dfrac{1}{3x + 4}\)

    Exercise \(\PageIndex{45}\)

    \(\dfrac{20y^2 + 15y - 4}{4y + 3}\)

    Exercise \(\PageIndex{46}\)

    \(\dfrac{4x^3 + 4x^2 - 3x - 2}{2x - 1}\)

    Answer

    \(2x^2 + 3x - \dfrac{2}{2x - 1}\)

    Exercise \(\PageIndex{47}\)

    \(\dfrac{9a^3 - 18a^2 8a - 1}{3a - 2}\)

    Exercise \(\PageIndex{48}\)

    \(\dfrac{4x^4 - 4x^3 + 2x^2 - 2x - 1}{x-1}\)

    Answer

    \(4x^3 + 2x - \dfrac{1}{x-1}\)

    Exercise \(\PageIndex{49}\)

    \(\dfrac{3y^4 + 9y^3 - 2y^2 - 6y + 4}{y + 3}\)

    Exercise \(\PageIndex{50}\)

    \(\dfrac{3y^2 + 3y + 5}{y^2 + y + 1}\)

    Answer

    \(3 + \dfrac{2}{y^2 + y + 1}\)

    Exercise \(\PageIndex{51}\)

    \(\dfrac{2a^2 + 4a + 1}{a^2 + 2a + 3}\)

    Exercise \(\PageIndex{52}\)

    \(\dfrac{8z^6 - 4z^5 - 8z^4 + 8z^3 + 3z^2 - 14z}{2z - 3}\)

    Answer

    \(4z^5 + 4z^4 + 2z^3 + 7z^2 + 12z + 11 + \dfrac{33}{2z - 3}\)

    Exercise \(\PageIndex{53}\)

    \(\dfrac{9 a^{7}+15 a^{6}+4 a^{5}-3 a^{4}-a^{3}+12 a^{2}+a-5}{3 a+1}\)

    Exercise \(\PageIndex{54}\)

    \((2x^5 + 5x^4 -1) \div (2x + 5)\)

    Answer

    \(x^4 - \dfrac{1}{2x + 5}\)

    Exercise \(\PageIndex{55}\)

    \((6a^4 - 2a^3 - 3a^2 + a + 4) \div (3a - 1)\)

    Exercises For Review

    Exercise \(\PageIndex{56}\)

    Find the product. \(\dfrac{x^2 + 2x - 8}{x^2 - 9} \cdot \dfrac{2x + 6}{4x - 8}\)

    Answer

    \(\dfrac{x + 4}{2(x-3)}\)

    Exercise \(\PageIndex{57}\)

    Find the sum. \(\dfrac{x-7}{x + 5} + \dfrac{x + 4}{x - 2}\)

    Exercise \(\PageIndex{58}\)

    Solve the equation \(\dfrac{1}{x + 3} + \dfrac{1}{x - 3} = \dfrac{1}{x^2 - 9}\)

    Answer

    \(x = \dfrac{1}{2}\)

    Exercise \(\PageIndex{59}\)

    When the same number is subtracted from both the numerator and denominator of \(dfrac{3}{10}\), the result is \(\dfrac{1}{8}\). What is teh number that is subtracted?

    Exercise \(\PageIndex{60}\)

    Simplify \(\dfrac{\frac{1}{x+5}}{\frac{4}{x^{2}-25}}\)

    Answer

    \(\dfrac{x-5}{4}\)


    This page titled 8.10: Dividing Polynomials is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) .

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