Chapter 6: Polynomials

6.7 Dividing Polynomials

Dividing polynomials is a process very similar to long division of whole numbers. But before looking at that, first master dividing a polynomial by a monomial. The way to do this is very similar to distributing, but the operation to distribute is the division, dividing each term by the monomial and reducing the resulting expression. This is shown in the following examples.

Example 6.7.1

Divide the following:

  1. (9x^5 + 6x^4 - 18x^3 - 24x^2)\div 3x^2Breaking this up into fractions, we get:\dfrac{9x^5}{3x^2}+ \dfrac{6x^4}{3x^2}- \dfrac{18x^3}{3x^2}- \dfrac{24x^2}{3x^2}Which yields:3x^3+2x^2-6x-8
  2. (8x^3 + 4x^2 - 2x + 6)\div 4x^2Breaking this up into fractions, we get:\dfrac{8x^3}{4x^2}+ \dfrac{4x^2}{4x^2} -\dfrac{2x}{4x^2} +\dfrac{6}{4x^2}Which yields:2x+1-\dfrac{1}{2x}+\dfrac{3}{2x^2}

Long division is required when dividing by more than just a monomial. Long division with polynomials is similar to long division with whole numbers.

Example 6.7.2

Divide the polynomial 3x^3 - 5x^2 - 32x + 7 by x - 4.

\polylongdiv{3x^3 - 5x^2 - 32x + 7}{x - 4}

The steps to get this result are as follows:

  1. Divide 3x^3 by x, yielding 3x^2. Multiply (x-4) by 3x^2, yielding 3x^3+12x^2. Subtract and bring down the next term and repeat.
  2. Divide 7x^2 by x, yielding 7x. Multiply (x-4) by 7x, yielding 7x^2-28x. Subtract and bring down the next term and repeat.
  3. Divide -4x by x, yielding -4. Multiply (x-4) by -4, yielding -4x+16. Subtract.

The solution can be written as either 3x^2+7x-4 \text{ R }-9 or 3x^2+7x-4-\dfrac{9}{x-4}.

The more formal way of writing this answer is the second option.

Example 6.7.3

Divide the polynomial 6x^3 - 8x^2 + 10x + 100 by 2x + 4.

\polylongdiv{6x^3 - 8x^2 + 10x + 100}{2x + 4}

The steps to get this result are as follows:

  1. Divide 6x^3 by 2x, yielding 3x^2. Multiply (2x+4) by 3x^2, yielding 6x^3+12x^2. Subtract and bring down the next term and repeat.
  2. Divide -20x^2 by 2x, yielding -10x. Multiply (2x+4) by -10x, yielding -20x^2-40x. Subtract and bring down the next term and repeat.
  3. Divide 50x by 2x, yielding 25. Multiply (2x+4) by 25, yielding 50x+100. Subtract.

The solution is 3x^2  - 10x  + 25 with no remainder.

Questions

Solve the following polynomial divisions.

  1. (20x^4 + x^3 + 2x^2)\div (4x^3)
  2. (5x^4 + 45x^3 + 4x^2) \div (9x)
  3. (20n^4 + n^3 + 40n^2) \div (10n)
  4. (3k^3 + 4k^2 + 2k) \div (8k)
  5. (12x^4 + 24x^3 + 3x^2) \div (6x)
  6. (5p^4 + 16p^3 + 16p^2) \div (4p)
  7. (10n^4 + 50n^3 + 2n^2) \div (10n^2)
  8. (3m^4 + 18m^3 + 27m^2) \div (9m^2)
  9. (45x^2 + 56x + 16) \div (9x + 4)
  10. (6x^2 + 16x + 16) \div (6x - 2)
  11. (10x^2 - 32x + 6) \div (10x - 2)
  12. (x^2 + 7x + 12) \div (x + 4)
  13. (4x^2 - 33x + 35) \div (4x - 5)
  14. (4x^2 - 23x - 35) \div (4x + 5)
  15. (x^3 + 15x^2 + 49x - 49) \div (x + 7)
  16. (6x^3 - 12x^2 - 43x - 20) \div (x - 4)
  17. (x^3 - 6x - 40) \div (x + 4)
  18. (x^3 - 16x^2 + 512) \div (x - 8)
  19. (x^3 - x^2 - 8x - 16) \div (x - 4)
  20. (2x^3 + 6x^2 + 4x + 12) \div (2x + 6)
  21. (12x^3 + 12x^2 - 15x - 9) \div (2x + 3)
  22. (6x + 18 - 21x^2 + 4x^3) \div (4x + 3)

<a class=”internal” href=”/intermediatealgebraberg/back-matter/answer-key-6-7/”>Answer Key 6.7

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