Polynomial Arithmetic

  1. Add or subtract the polynomials given, as indicated:

    1. $\displaystyle{(3y^2+5y-7)+(2y^2-7y+10)}$   $\ans{\displaystyle{5y^2-2y+3}}$

    2. $\displaystyle{(6n^2 - 3n + 4) + (3n^3 - n^2)}$   $\ans{\displaystyle{3n^3+5n^2-3n+4}}$

    3. $\displaystyle{(4x^3+3x^2+2x+1) + (x^3 + 2x^2 + 3x + 4)}$   $\ans{\displaystyle{5x^3+5x^2+5x+5}}$

    4. $\displaystyle{(x^3-3x^2y+4xy^2+y^3)+(7x^3+x^2y-9xy^2+y^3)}$   $\ans{\displaystyle{8x^3-2x^2y-5xy^2+2y^3}}$

    5. $\displaystyle{(7m^2+9m+3) \, - \, (3m^2 + 8m + 2)}$   $\ans{\displaystyle{4m^2+m+1}}$

    6. $\displaystyle{(4x+11)-(3x^2+7x-3)}$   $\ans{\displaystyle{-3x^2-3x+14}}$

    7. $\displaystyle{(5xy^4 - 7xy^2 + 4x^2 - 3) \, - \, (-3xy^4 + 2xy^2 - 2y + 4)}$   $\ans{\displaystyle{8xy^4 - 9xy^2 + 4x^2 + 2y - 7}}$

    8. $\displaystyle{(2x^3-5x^2y+xy^2-y^3)-(8x^3-x^2y-3xy^2+y^3)}$   $\ans{\displaystyle{-6x^3-4x^2y+4xy^2-2y^3}}$

  2. Multiply:

    1. $\displaystyle{(3x+2)(5x-7)}$   $\ans{\displaystyle{15x^2-11x-14}}$

    2. $\displaystyle{(x-7)(-2x+5)}$   $\ans{\displaystyle{-2x^2+19x-35}}$

    3. $\displaystyle{(3x+y)(7x-3y)}$   $\ans{\displaystyle{21x^2-2xy-3y^2}}$

    4. $\displaystyle{(11x-3)(11x+3)}$   $\ans{\displaystyle{121x^2-9}}$

    5. $\displaystyle{(2b+1)(2b-1)(3b-4)}$   $\ans{\displaystyle{12b^3-16b^2-3b+4}}$

    6. $\displaystyle{(5w-3)^2}$   $\ans{\displaystyle{25w^2-30w+9}}$

    7. $\displaystyle{(2q+3)^3}$   $\ans{\displaystyle{8q^3+36q^2+54q+27}}$

    8. $\displaystyle{(x+1)^4}$   $\ans{\displaystyle{x^4+4x^3+6x^2+4x+1}}$

    9. $\displaystyle{(2x-3)^4}$   $\ans{\displaystyle{16x^4-96x^3+216x^2-216x+81}}$

    10. $\displaystyle{(2x^3 - 5y)^2}$   $\ans{\displaystyle{4x^6 - 20x^3 y + 25y^2}}$

    11. $\displaystyle{(x+t)(x^2-xt+t^2)}$   $\ans{\displaystyle{x^3 + t^3}}$

    12. $\displaystyle{(5a + 4b)^3}$   $\ans{\displaystyle{125a^3 + 300a^2b + 240ab^2 + 64b^3}}$

    13. $\displaystyle{3x(4xy^3-7x^2y-3y)}$   $\ans{\displaystyle{12x^2y^3-21x^3y-9xy}}$

    14. $\displaystyle{(2a-3b)(3a+4ab+b)}$   $\ans{\displaystyle{6a^2+8a^2b-7ab-12ab^2-3b^2}}$

    15. $\displaystyle{(y+4)(y^2-7y+12)}$   $\ans{\displaystyle{y^3-3y^2-16y+48}}$

    16. $\displaystyle{(x^2+2x-3)(5x^2+3x-7)}$   $\ans{\displaystyle{5x^4+13x^3-16x^2-23x+21}}$

    17. $\displaystyle{(4m^2-m+8)(m^3+2m^2+3m+4)}$   $\ans{\displaystyle{4m^5+7m^4+18m^3+29m^2+20m+32}}$

  3. Expand and Simplify:

    1. $\displaystyle{(x+1)^3+(x+1)^2+(x+1)+1}$   $\ans{\displaystyle{x^3+4x^2+6x+4}}$

    2. $\displaystyle{(2-x+3y)(2-x-3y)(4x^3-3+2x) + (5x-3)^3}$   $\ans{\displaystyle{4 x^5-16 x^4-36 x^3 y^2+143 x^3-236 x^2-18 x y^2+155 x+27 y^2-39}}$