Honors Algebra 2: Extra Practice on Operations on Polynomials with Constant Exponents

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In the following problems, assume no denominator equals zero.

  1. \( \displaystyle 3x^{3a} + 24y^{3b} \)
    2. Solution

    \( \displaystyle 3\left( x^a + 2y^b \right) \left( x^{2a} - 2x^ay^b + 4y^{2b} \right) \)

  2. Factor: \( \displaystyle \frac{1}{16}x^{3a} + \frac{1}{2}y^{6a}z^{9b} \)
    3. Solution

    \( \frac{1}{2} \left( \frac{1}{2}x^a + y^{2a}z^{3b} \right) \left( \frac{1}{4} x^{2a} - \frac{1}{2}x^ay^{2a}z^{3b} + y^{4a}z^{6b} \right) \)

  3. Factor: \( \displaystyle c^{2w+1} + 2c^{w+1} \)
    4. Solution

    \( \displaystyle c^{w+1}\left( c^w + 2 \right) \)

  4. Simplify: \( \displaystyle \frac{x^{4n+1}+x^{5n}}{x^{6n} - x^{5n+2}} \)
    5. Solution

    \( \displaystyle \frac{x + x^n}{x^n (x^n - x^2)} \)

  5. Simplify: \( \displaystyle \frac{x^{4n} + 7x^{2n} + 12}{x^{3n} + 5x^{2n} + 4x^n + 20} \)
    6. Solution

    \( \displaystyle \frac{x^{2n} + 3}{x^n + 5} \)

  6. Simplify: \( \displaystyle \frac{x^{6n} - y^{3n}}{x^{4n} - y^{2n}} \)
    7. Solution

    \( \displaystyle \frac{x^{4n} + x^{2n}y^n + y^{2n}}{x^{2n} + y^n} \)

  7. Simplify: \( \displaystyle \frac{x^{2a} - y^{2b}}{4x^{2c} - 20x^c + 25} \cdot \frac{8x^c - 20}{3x^a + 3y^b} \)
    8. Solution

    \( \displaystyle \frac{4(x^a - y^b)}{3(2x^c - 5)} \)

  8. Simplify: \( \displaystyle \frac{x^{3a} + x^{2a+1}}{x^{b+c}} \div \frac{x^{3a} - x^{a + 2}}{x^{a + 2b} - x^{2b + 1}} \)
    9. Solution

    \( \displaystyle \frac{x^ax^b}{x^c} \)

  9. Simplify: \( \displaystyle \frac{20-x^c}{x^{2c} + 4x^c} + \frac{x^c + 10}{x^c + 4}\)
    10. Solution

    \( \displaystyle \frac{x^c + 5}{x^c} \)

  10. Simplify: \( \displaystyle \frac{y^n}{y^n+6} - \frac{6 - 5y^n}{y^{2n} + 6y^n} \)
    11. Solution

    \( \displaystyle \frac{y^n-1}{y^n} \)

  11. Simplify: \( \displaystyle \frac{1}{x^{3a}y^{5n}} + \frac{1}{x^{2a+4}y^{n+3}} \)
    12. Solution

    \( \displaystyle \frac{x^4y^3 + x^ay^{4n}}{x^{3n+4}y^{5n+3}} \)

  12. \( \displaystyle \frac{5y^b}{6x^c} - \frac{2}{9x^cy^b} + \frac{4x^c}{3y^{2b}} \)
    13. Solution

    \( \displaystyle \frac{15y^{3b} - 4y^b + 24x^{2c}}{18x^cy^{2b}} \)

  13. \( \displaystyle \frac{x^{2n+1} + 2x^{n+1} - 3xy^{2n}}{x^{2n+1} + 5x^{n+1}y^n + 6xy^{2n}} \)
    14. Solution

    \( \displaystyle \frac{x^n-y^n}{x^n + 2y^n} \)