Answer Key 6.5

  1. 6p-42
  2. 32k^2+16k
  3. 12x+6
  4. 18n^3+21n^2
  5. \begin{array}{rrrrrr} \\ \\ \\ \\ \\ &8n&+&8&& \\ \times &4n&+&6&& \\ \midrule &32n^2&+&32n&& \\ &&&48n&+&48 \\ \midrule &32n^2&+&80n&+&48 \end{array}
  6. \begin{array}{rrrrrr} \\ \\ \\ \\ \\ &x&-&4&& \\ \times &2x&+&1&& \\ \midrule &2x^2&-&8x&& \\ &&&x&-&4 \\ \midrule &2x^2&-&7x&-&4 \end{array}
  7. \begin{array}{rrrrrr} \\ \\ \\ \\ \\ &7b&-&5&& \\ \times &8b&+&3&& \\ \midrule &56b^2&-&40b&& \\ &&+&21b&-&15 \\ \midrule &56b^2&-&19b&-&15 \end{array}
  8. \begin{array}{rrrrrr} \\ \\ \\ \\ \\ &4r&+&8&& \\ \times &r&+&8&& \\ \midrule &4r^2&+&8r&& \\ &&+&32r&+&64 \\ \midrule &4r^2&+&40r&+&64 \end{array}
  9. \begin{array}{rrrrrr} \\ \\ \\ \\ \\ &5v&-&2&& \\ \times &3v&-&4&& \\ \midrule &15v^2&-&6v&& \\ &&-&20v&+&8 \\ \midrule &15v^2&-&26v&+&8 \end{array}
  10. \begin{array}{rrrrrr} \\ \\ \\ \\ \\ &a&-&8&& \\ \times &6a&+&4&& \\ \midrule &6a^2&-&48a&& \\ &&+&4a&-&32 \\ \midrule &6a^2&-&44a&-&32 \end{array}
  11. \begin{array}{rrrrrr} \\ \\ \\ \\ \\ &6x&-&4y&& \\ \times &5x&+&y&& \\ \midrule &30x^2&-&20xy&& \\ &&+&6xy&-&4y^2 \\ \midrule &30x^2&-&14xy&-&4y^2 \end{array}
  12. \begin{array}{rrrrrr} \\ \\ \\ \\ \\ &8u&-&7v&& \\ \times &2u&+&3v&& \\ \midrule &16u^2&-&14uv&& \\ &&+&24uv&-&21v^2 \\ \midrule &16u^2&+&10uv&-&21v^2 \end{array}
  13. \begin{array}{rrrrrr} \\ \\ \\ \\ \\ &8x&+&3y&& \\ \times &7x&+&5y&& \\ \midrule &56x^2&+&21xy&& \\ &&+&40xy&+&15y^2 \\ \midrule &56x^2&+&61xy&+&15y^2 \end{array}
  14. \begin{array}{rrrrrr} \\ \\ \\ \\ \\ &a&-&3b&& \\ \times &5a&+&8b&& \\ \midrule &5a^2&-&15ab&& \\ &&+&8ab&-&24b^2 \\ \midrule &5a^2&-&7ab&-&24b^2 \end{array}
  15. \begin{array}{rrrrrrrr} \\ \\ \\ \\ \\ &6r^2&-&r&+&5&& \\ \times &&&r&-&7&& \\ \midrule &6r^3&-&r^2&+&5r&& \\ &&-&42r^2&+&7r&-&35 \\ \midrule &6r^3&-&43r^2&+&12r&-&35 \end{array}
  16. \begin{array}{rrrrrrrr} \\ \\ \\ \\ \\ &4x^2&+&3x&+&5&& \\ \times &&&4x&+&8&& \\ \midrule &16x^3&+&12x^2&+&20x&& \\ &&+&32x^2&+&24x&+&40 \\ \midrule &16x^3&+&44x^2&+&44x&+&40 \end{array}
  17. \begin{array}{rrrrrrrr} \\ \\ \\ \\ \\ &2n^2&-&2n&+&5&& \\ \times &&&6n&-&4&& \\ \midrule &12n^3&-&12n^2&+&30n&& \\ &&-&8n^2&+&8n&-&20 \\ \midrule &12n^3&-&20n^2&+&38n&-&20 \end{array}
  18. \begin{array}{rrrrrrrr} \\ \\ \\ \\ \\ &4b^2&+&4b&+&4&& \\ \times &&&2b&-&3&& \\ \midrule &8b^3&+&8b^2&+&8b&& \\ &&-&12b^2&-&12b&-&12 \\ \midrule &8b^3&-&4b^2&-&4b&-&12 \end{array}
  19. \begin{array}{rrrrrrrr} \\ \\ \\ \\ \\ &6x^2&-&7xy&+&4y^2&& \\ \times &&&6x&+&3y&& \\ \midrule &36x^3&-&42x^2y&+&24xy^2&& \\ &&+&18x^2y&-&21xy^2&+&12y^3 \\ \midrule &36x^3&-&24x^2y&+&3xy^2&+&12y^3 \end{array}
  20. \begin{array}{rrrrrrrr} \\ \\ \\ \\ \\ &7m^2&+&6mn&+&4n^2&& \\ \times &&&3m&-&2n&& \\ \midrule &21m^3&+&18m^2n&+&12mn^2&& \\ &&-&14m^2n&-&12mn^2&-&8n^3 \\ \midrule &21m^3&+&4m^2n&-&8n^3&& \end{array}
  21. \begin{array}{rrrrrrrrrr} \\ \\ \\ \\ \\ \\ &6n^2&-&5n&+&6&&&& \\ \times&8n^2&+&4n&+&6&&&& \\ \midrule &48n^4&-&40n^3&+&48n^2&&&& \\ &&+&24n^3&-&20n^2&+&24n&& \\ &&&&+&36n^2&-&30n&+&36 \\ \midrule &48n^4&-&16n^3&+&64n^2&-&6n&+&36 \end{array}
  22. \begin{array}{rrrrrrrrrr} \\ \\ \\ \\ \\ \\ &7a^2&-&6a&+&1&&&& \\ \times&2a^2&+&6a&+&3&&&& \\ \midrule &14a^4&-&12a^3&+&2a^2&&&& \\ &&+&42a^3&-&36a^2&+&6a&& \\ &&&&+&21a^2&-&18a&+&3 \\ \midrule &14a^4&+&30a^3&-&13a^2&-&12a&+&3 \end{array}
  23. \begin{array}{rrrrrrrrrr} \\ \\ \\ \\ \\ \\ &3k^2&+&3k&+&6&&&& \\ \times&5k^2&+&3k&+&3&&&& \\ \midrule &15k^4&+&15k^3&+&30k^2&&&& \\ &&+&9k^3&+&9k^2&+&18k&& \\ &&&&+&9k^2&+&9k&+&18 \\ \midrule &15k^4&+&24k^3&+&48k^2&+&27k&+&18 \end{array}
  24. \begin{array}{rrrrrrrrrr} \\ \\ \\ \\ \\ \\ &6u^2&+&4uv&+&3v^2&&&& \\ \times &7u^2&+&8uv&-&6v^2&&&& \\ \midrule &42u^4&+&28u^3v&+&21u^2v^2&&&& \\ &&+&48u^3v&+&32u^2v^2&+&24uv^3&& \\ &&&&-&36u^2v^2&-&24uv^3&-&18v^4 \\ \midrule &42u^4&+&76u^3v&+&17u^2v^2&-&18v^4&& \\ \end{array}
  25. \begin{array}{rrrrrrrrrrrrrr} \\ \\ \\ \\ \\ \\ \\ &2n^3&-&8n^2&+&3n&+&6&&&&&& \\ \times&n^3&-&6n^2&-&2n&+&3&&&&&& \\ \midrule &2n^6&-&8n^5&+&3n^4&+&6n^3&&&&&& \\ &&-&12n^5&+&48n^4&-&18n^3&-&36n^2&&&& \\ &&&&-&4n^4&+&16n^3&-&6n^2&-&12n&& \\ &&&&&&+&6n^3&-&24n^2&+&9n&+&18 \\ \midrule &2n^6&-&20n^5&+&47n^4&+&10n^3&-&66n^2&-&3n&+&18 \end{array}
  26. \begin{array}{rrrrrrrrrrrrrr} \\ \\ \\ \\ \\ \\ \\ &a^3&+&2a^2&+&3a&+&3&&&&&& \\ \times&a^3&+&2a^2&-&4a&+&1&&&&&& \\ \midrule &a^6&+&2a^5&+&3a^4&+&3a^3&&&&&& \\ &&+&2a^5&+&4a^4&+&6a^3&+&6a^2&&&& \\ &&&&-&4a^4&-&8a^3&-&12a^2&-&12a&& \\ &&&&&&+&a^3&+&2a^2&+&3a&+&3 \\ \midrule &a^6&+&4a^5&+&3a^4&+&2a^3&-&4a^2&-&9a&+&3 \end{array}
  27. \begin{array}{rrrrrr} \\ \\ \\ \\ \\ \\ &2x&+&1&& \\ \times &3x&-&4&& \\ \midrule &6x^2&+&3x&& \\ &&-&8x&-&4 \\ \midrule &3(6x^2&-&5x&-&4) \\ &18x^2&-&15x&-&12 \end{array}
  28. \begin{array}{rrrrrr} \\ \\ \\ \\ \\ \\ &2x&-&3&& \\ \times &x&-&4&& \\ \midrule &2x^2&-&3x&& \\ &&-&8x&+&12 \\ \midrule &5(2x^2&-&11x&+&12) \\ &10x^2&-&55x&+&60 \end{array}
  29. \begin{array}{rrrrrr} \\ \\ \\ \\ \\ \\ &4x&-&5&& \\ \times &2x&+&1&& \\ \midrule &8x^2&-&10x&& \\ &&+&4x&-&5 \\ \midrule &3(8x^2&-&6x&-&5) \\ &24x^2&-&18x&-&15 \end{array}
  30. \begin{array}{rrrrrr} \\ \\ \\ \\ \\ \\ &2x&-&6&& \\ \times &4x&+&1&& \\ \midrule &8x^2&-&24x&& \\ &&+&2x&-&6 \\ \midrule &2(8x^2&-&22x&-&6) \\ &16x^2&-&44x&-&12 \end{array}

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