Dividing Polynomials Notes

Dividing Polynomials

Dividing a Polynomial by a Monomial

  • Divide each term of the polynomial by the monomial.
  • Simplify each term after division.

Example 1

  • Divide (2x^3 + 4x^2 - 2x + 3) by 2x.
  • \frac{2x^3}{2x} + \frac{4x^2}{2x} - \frac{2x}{2x} + \frac{3}{2x}
  • Simplify each term:
    • \frac{2x^3}{2x} = x^2
    • \frac{4x^2}{2x} = 2x
    • \frac{2x}{2x} = 1
    • \frac{3}{2x} = \frac{3}{2x}
  • Final answer: x^2 + 2x - 1 + \frac{3}{2x}

Example 2

  • Divide (6w^4 + 4w^3 - w^2) by 2w^3.
  • \frac{6w^4}{2w^3} + \frac{4w^3}{2w^3} - \frac{w^2}{2w^3}
  • Simplify each term:
    • \frac{6w^4}{2w^3} = 3w
    • \frac{4w^3}{2w^3} = 2
    • \frac{w^2}{2w^3} = \frac{1}{2w}
  • Final answer: 3w + 2 - \frac{1}{2w}

Dividing Polynomials by Binomials

  • Similar to long division learned in elementary school.
  • Express remainders as a fraction with the divisor as the denominator.

Example: Long Division Analogy

  • Divide 643 by 7.
  • 7 goes into 64 nine times (9 * 7 = 63).
  • 64 - 63 = 1, bring down the 3 to get 13.
  • 7 goes into 13 one time (1 * 7 = 7).
  • 13 - 7 = 6.
  • Remainder is 6, expressed as \frac{6}{7}.
  • Final answer: 91 + \frac{6}{7}.

Example 1

  • Divide (d^2 + 3d - 7) by (d - 2).

  • Set up the long division:

             d + 5
d - 2 | d^2 + 3d - 7
       -(d^2 - 2d)
         5d - 7
       -(5d - 10)
              3

  • d times what equals d^2? Answer: d. Multiply d by (d - 2) to get (d^2 - 2d).

  • Subtract (d^2 - 2d) from (d^2 + 3d) to get 5d. Bring down the -7.

  • d times what equals 5d? Answer: 5. Multiply 5 by (d - 2) to get (5d - 10).

  • Subtract (5d - 10) from (5d - 7) to get 3.

  • Remainder is 3, expressed as \frac{3}{d-2}.

  • Final answer: d + 5 + \frac{3}{d-2}.

Example 2: Reordering Polynomials

  • Divide (3y - 11 + 2y^2) by (y - 3).

  • Reorder the polynomial to descending order: 2y^2 + 3y - 11.

  • Set up the long division:

             2y + 9
y - 3 | 2y^2 + 3y - 11
       -(2y^2 - 6y)
             9y - 11
       -(9y - 27)
                  16

  • y times what equals 2y^2? Answer: 2y. Multiply 2y by (y - 3) to get (2y^2 - 6y).

  • Subtract (2y^2 - 6y) from (2y^2 + 3y) to get 9y. Bring down the -11.

  • y times what equals 9y? Answer: 9. Multiply 9 by (y - 3) to get (9y - 27).

  • Subtract (9y - 27) from (9y - 11) to get 16.

  • Remainder is 16, expressed as \frac{16}{y-3}.

  • Final answer: 2y + 9 + \frac{16}{y-3}.

Example 3: Placeholders and Reordering

  • Divide (12a^3 + 5a - 6) by (2a - 1).

  • Notice that the polynomial is missing the a^2 term. Add a placeholder.

  • Rewrite the polynomial with the placeholder: 12a^3 + 0a^2 + 5a - 6.

  • Set up the long division:

                 6a^2 + 3a + 4
2a - 1 | 12a^3 + 0a^2 + 5a - 6
         -(12a^3 - 6a^2)
                  6a^2 + 5a
         -(6a^2 - 3a)
                       8a - 6
         -(8a - 4)
                        -2

  • 2a times what equals 12a^3? Answer: 6a^2. Multiply 6a^2 by (2a - 1) to get (12a^3 - 6a^2).

  • Subtract (12a^3 - 6a^2) from (12a^3 + 0a^2) to get 6a^2. Bring down the 5a.

  • 2a times what equals 6a^2? Answer: 3a. Multiply 3a by (2a - 1) to get (6a^2 - 3a).

  • Subtract (6a^2 - 3a) from (6a^2 + 5a) to get 8a. Bring down the -6.

  • 2a times what equals 8a? Answer: 4. Multiply 4 by (2a - 1) to get (8a - 4).

  • Subtract (8a - 4) from (8a - 6) to get -2.

  • Remainder is -2, expressed as \frac{-2}{2a-1}.

  • Final answer: 6a^2 + 3a + 4 - \frac{2}{2a-1}.