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Simplify the following algebraic expressions: 1. (4x+2)/2 2. (9a2-3)/(-3) 3. (8y2+4y-1)/4 4. (6a-4b+2c)/(-2) 5. (5x2-10x)/x, x!=0

Mathematics

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1. Simplify 4x+22\frac{4x+2}{2}

  1. Divide each term in the numerator by the denominator.

    4x+22=4x2+22\frac{4x+2}{2} = \frac{4x}{2} + \frac{2}{2}

  2. Simplify each term.

    =2x+1= 2x + 1

    The simplified expression is 2x+1\boxed{2x+1}.

    2. Simplify 9a2−3−3\frac{9a^2-3}{-3}

  3. Divide each term in the numerator by the denominator.

    9a2−3−3=9a2−3−3−3\frac{9a^2-3}{-3} = \frac{9a^2}{-3} - \frac{3}{-3}

  4. Simplify each term.

    =−3a2−(−1)= -3a^2 - (-1) =−3a2+1= -3a^2 + 1

    The simplified expression is −3a2+1\boxed{-3a^2+1}.

    3. Simplify 8y2+4y−14\frac{8y^2+4y-1}{4}

  5. Divide each term in the numerator by the denominator.

    8y2+4y−14=8y24+4y4−14\frac{8y^2+4y-1}{4} = \frac{8y^2}{4} + \frac{4y}{4} - \frac{1}{4}

  6. Simplify each term.

    =2y2+y−14= 2y^2 + y - \frac{1}{4}

    The simplified expression is 2y2+y−14\boxed{2y^2+y-\frac{1}{4}}.

    4. Simplify 6a−4b+2c−2\frac{6a-4b+2c}{-2}

  7. Divide each term in the numerator by the denominator.

    6a−4b+2c−2=6a−2−4b−2+2c−2\frac{6a-4b+2c}{-2} = \frac{6a}{-2} - \frac{4b}{-2} + \frac{2c}{-2}

  8. Simplify each term.

    =−3a−(−2b)+(−c)= -3a - (-2b) + (-c) =−3a+2b−c= -3a + 2b - c

    The simplified expression is −3a+2b−c\boxed{-3a+2b-c}.

    5. Simplify 5x2−10xx\frac{5x^2-10x}{x}, x≠0x \neq 0

  9. Divide each term in the numerator by the denominator.

    5x2−10xx=5x2x−10xx\frac{5x^2-10x}{x} = \frac{5x^2}{x} - \frac{10x}{x}

  10. Simplify each term.

    =5x−10= 5x - 10

    The simplified expression is 5x−10\boxed{5x-10}.

    6. Simplify 4a4+a2−a2\frac{4a^4+a^2}{-a^2}, a≠0a \neq 0

  11. Divide each term in the numerator by the denominator.

    4a4+a2−a2=4a4−a2+a2−a2\frac{4a^4+a^2}{-a^2} = \frac{4a^4}{-a^2} + \frac{a^2}{-a^2}

  12. Simplify each term.

    =−4a4−2+(−1)= -4a^{4-2} + (-1) =−4a2−1= -4a^2 - 1

    The simplified expression is −4a2−1\boxed{-4a^2-1}.

    7. Simplify 3x3−6x2−3x−3x\frac{3x^3-6x^2-3x}{-3x}, x≠0x \neq 0

  13. Divide each term in the numerator by the denominator.

    3x3−6x2−3x−3x=3x3−3x−6x2−3x−3x−3x\frac{3x^3-6x^2-3x}{-3x} = \frac{3x^3}{-3x} - \frac{6x^2}{-3x} - \frac{3x}{-3x}

  14. Simplify each term.

    =−x3−1−(−2x2−1)−(−1)= -x^{3-1} - (-2x^{2-1}) - (-1) =−x2+2x+1= -x^2 + 2x + 1

    The simplified expression is −x2+2x+1\boxed{-x^2+2x+1}.

    8. Simplify −5m4−10m2−15m25m\frac{-5m^4-10m^2-15m^2}{5m}, m≠0m \neq 0

  15. Combine like terms in the numerator.

    −5m4−10m2−15m25m=−5m4−25m25m\frac{-5m^4-10m^2-15m^2}{5m} = \frac{-5m^4-25m^2}{5m}

  16. Divide each term in the numerator by the denominator.

    =−5m45m−25m25m= \frac{-5m^4}{5m} - \frac{25m^2}{5m}

  17. Simplify each term.

    =−m4−1−5m2−1= -m^{4-1} - 5m^{2-1} =−m3−5m= -m^3 - 5m

    The simplified expression is −m3−5m\boxed{-m^3-5m}.

    9. Simplify 7y3−4y2y2\frac{7y^3-4y^2}{y^2}, y≠0y \neq 0

  18. Divide each term in the numerator by the denominator.

    7y3−4y2y2=7y3y2−4y2y2\frac{7y^3-4y^2}{y^2} = \frac{7y^3}{y^2} - \frac{4y^2}{y^2}

  19. Simplify each term.

    =7y3−2−4y2−2= 7y^{3-2} - 4y^{2-2} =7y−4y0= 7y - 4y^0 =7y−4= 7y - 4

    The simplified expression is 7y−4\boxed{7y-4}.

    10. Simplify −16a4+8a3−4a2−4a2\frac{-16a^4+8a^3-4a^2}{-4a^2}, a≠0a \neq 0

  20. Divide each term in the numerator by the denominator.

    −16a4+8a3−4a2−4a2=−16a4−4a2+8a3−4a2−4a2−4a2\frac{-16a^4+8a^3-4a^2}{-4a^2} = \frac{-16a^4}{-4a^2} + \frac{8a^3}{-4a^2} - \frac{4a^2}{-4a^2}

  21. Simplify each term.

    =4a4−2−2a3−2−(−1)= 4a^{4-2} - 2a^{3-2} - (-1) =4a2−2a+1= 4a^2 - 2a + 1

    The simplified expression is 4a2−2a+1\boxed{4a^2-2a+1}.

    11. Simplify 15b3−5b2+105b2\frac{15b^3-5b^2+10}{5b^2}, b≠0b \neq 0

  22. Divide each term in the numerator by the denominator.

    15b3−5b2+105b2=15b35b2−5b25b2+105b2\frac{15b^3-5b^2+10}{5b^2} = \frac{15b^3}{5b^2} - \frac{5b^2}{5b^2} + \frac{10}{5b^2}

  23. Simplify each term.

    =3b3−2−1+2b2= 3b^{3-2} - 1 + \frac{2}{b^2} =3b−1+2b2= 3b - 1 + \frac{2}{b^2}

    The simplified expression is 3b−1+2b2\boxed{3b-1+\frac{2}{b^2}}.

    12. Simplify 24x3y−18x2y2+30xy36xy\frac{24x^3y-18x^2y^2+30xy^3}{6xy}, x,y≠0x, y \neq 0

  24. Divide each term in the numerator by the denominator.

    24x3y−18x2y2+30xy36xy=24x3y6xy−18x2y26xy+30xy36xy\frac{24x^3y-18x^2y^2+30xy^3}{6xy} = \frac{24x^3y}{6xy} - \frac{18x^2y^2}{6xy} + \frac{30xy^3}{6xy}

  25. Simplify each term.

    =4x3−1y1−1−3x2−1y2−1+5x1−1y3−1= 4x^{3-1}y^{1-1} - 3x^{2-1}y^{2-1} + 5x^{1-1}y^{3-1} =4x2y0−3xy+5x0y2= 4x^2y^0 - 3xy + 5x^0y^2 =4x2−3xy+5y2= 4x^2 - 3xy + 5y^2

    The simplified expression is 4x2−3xy+5y2\boxed{4x^2-3xy+5y^2}.

    13. Simplify −14m4n2+21m3n2−7m2n−7m2n\frac{-14m^4n^2+21m^3n^2-7m^2n}{-7m^2n}, m,n≠0m, n \neq 0

  26. Divide each term in the numerator by the denominator.

    −14m4n2+21m3n2−7m2n−7m2n=−14m4n2−7m2n+21m3n2−7m2n−7m2n−7m2n\frac{-14m^4n^2+21m^3n^2-7m^2n}{-7m^2n} = \frac{-14m^4n^2}{-7m^2n} + \frac{21m^3n^2}{-7m^2n} - \frac{7m^2n}{-7m^2n}

  27. Simplify each term.

    =2m4−2n2−1−3m3−2n2−1−(−1)= 2m^{4-2}n^{2-1} - 3m^{3-2}n^{2-1} - (-1) =2m2n−3mn+1= 2m^2n - 3mn + 1

    The simplified expression is 2m2n−3mn+1\boxed{2m^2n-3mn+1}.

    14. Simplify 18a4b4−27a3b3+33a2b2\frac{18a^4b^4-27a^3b^3+3}{3a^2b^2}, a,b≠0a, b \neq 0

  28. Divide each term in the numerator by the denominator.

    18a4b4−27a3b3+33a2b2=18a4b43a2b2−27a3b33a2b2+33a2b2\frac{18a^4b^4-27a^3b^3+3}{3a^2b^2} = \frac{18a^4b^4}{3a^2b^2} - \frac{27a^3b^3}{3a^2b^2} + \frac{3}{3a^2b^2}

  29. Simplify each term.

    =6a4−2b4−2−9a3−2b3−2+1a2b2= 6a^{4-2}b^{4-2} - 9a^{3-2}b^{3-2} + \frac{1}{a^2b^2} =6a2b2−9ab+1a2b2= 6a^2b^2 - 9ab + \frac{1}{a^2b^2}

    The simplified expression is 6a2b2−9ab+1a2b2\boxed{6a^2b^2-9ab+\frac{1}{a^2b^2}}.

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