Section 1.5
Multiplying Polynomials
When multiplying two binomials together, you must multiply each term of the first binomial by each term of the second binomial. That involves four multiplications—first term times first term, first term times second term, second times first, and second times second. This process is frequently referred to as the FOIL method--Firsts, Outers, Inners, Lasts.
The patterns illustrated in examples (c.) and (d.) below are particularly important — the "difference of two squares" and the perfect trinomial square, respectively.
When multiplying expressions with more than two terms, each of the terms in the first expression must be multiplied by each of the terms in the second expression (see examples (g.) and (h.) below).
Examples:
a.
\begin{align} (x+3)(x+7)&=x^2+7x+3x+3\cdot 7\\ &=x^2+10x+21 \end{align} |
b.
\begin{align} (x+2)(x-5)&=x^2-5x+2x-2\cdot 5\\ &=x^2-3x-10 \end{align} |
c.
\begin{align} (x+5)(x-5)&=x^2+5x-5x-5\cdot 5\\ &=x^2-25 \end{align} |
d.
\begin{align} (x-7)(x-7)&=x^2-7x-7x+7\cdot 7\\ &=x^2-14+49 \end{align} |
e.
\begin{align} (2x-5)(3x-7)&=6x^2-14x-15x+35\\ &=x^2-14x+49 \end{align} |
f.
\begin{align} (3x-5)(7x+2)&=21x^2+6x-35x-10\\ &=21x^2-29x-10 \end{align} |
g.
\begin{align} (x+3)(x^2-4x+5)&=x^3-4x^2+5x+3x^2-12x+15\\ &=x^3-x^2-7x+15 \end{align} The $x$ from the first factor is multiplied by each of the three terms in the second factor. Then the 4 from the first factor is multiplied by each of the terms in the second factor. |
|
h. \begin{align} (2x^2-3x+4)(3x^2+x-7)&=6x^4+2x^3-14x^2-9x^3-3x^2+21x+12x^2+4x-28\\ &=6x^4-7x^3-5x^2+25x-28 \end{align} Each of the three terms in the first factor is multiplied by each of the three terms in the second factor, producing (at first) nine terms in the product. These can then be combined to give the five-term final result. |
Exercises:
Determine each of the following products.
\begin{array}{l l l}
1. \;\;(x+2)(x+7)\;\; &2. \;\;(x+5)(x+1)\;\; &3. \;\;(x+8)(x+3)\;\;\\
4. \;\;(x-2)(x-4)\;\; &5. \;\;(x-5)(x-3)\;\; &6. \;\;(x-6)(x-4)\;\;\\
7. \;\;(x+2)(x-7)\;\; &8. \;\;(x-5)(x+1)\;\; &9. \;\;(x+8)(x-3)\;\;\\
10. \;\;(x-3)(x+7)\;\; &11. \;\;(x-4)(x-5)\;\; &12. \;\;(x+5)(x-4)\;\;\\
13. \;\;(x-3)(x-7)\;\; &14. \;\;(x+10)(x-4)\;\; &15. \;\;(x+12)(x-1)\;\;\\
16. \;\;(x+3)(x-11)\;\; &17. \;\;(x+3)(x-6)\;\; &18. \;\;(x+4)(x-10)\;\;\\
19. \;\;(x-7)(x+3)\;\; &20. \;\;(x+1)(x-20)\;\; &21. \;\;(x+2)(x+18)\;\;\\
22. \;\;(x-6)(x+5)\;\; &23. \;\;(x+3)(x-12)\;\; &24. \;\;(x-9)(x+7)\;\;\\
25. \;\;(x+7)(x-8)\;\; &26. \;\;(x+4)(x-3)\;\; &27. \;\;(x-6)(x+2)\;\;\\
28. \;\;(x+4)(x-7)\;\; &29. \;\;(x-2)(x+3)\;\; &30. \;\;(x-1)(x+6)\;\;\\
31. \;\;(x-1)(x+1)\;\; &32. \;\;(x+2)(x-2)\;\; &33. \;\;(x+6)(x-6)\;\;\\
34. \;\;(2x+3)(2x-3)\;\; &35. \;\;(x-4)(x+7)\;\; &36. \;\;(7x-1)(7x+1)\;\;\\
37. \;\;(x+5)(x-5)\;\; &38. \;\;(x-7)(x+7)\;\; &39. \;\;(x-9)(x+9)\;\;\\
40. \;\;(x-3)(x+3)\;\; &41. \;\;(2x-5)(x+2)\;\; &42. \;\;(4x+3)(3x-2)\;\;\\
43. \;\;(3x-5)(3x+5)\;\; &44. \;\;(2x+3)(2x-3)\;\; &45. \;\;(5x-7)(5x+7)\;\;\\
46. \;\;(x+4)(x-4)\;\; &47. \;\;\left( {{x}^{2}}-1 \right)\left( {{x}^{2}}+1 \right)\;\; &48. \;\;(2x+7)(2x-7)\;\;\\
49. \;\;(3x-1)(3x+1)\;\; &50. \;\;(7x-8)(7x+8)\;\; &51. \;\;(10-3x)(10+3x)\;\;\\
52. \;\;(5x-3y)(5x+3y)\;\; &53. \;\;(2t-5K)(2t+5K)\;\; &54. \;\;(3-7t)(3+7t)\;\;\\
55. \;\;{{(x-3)}^{2}}\;\; &56. \;\;{{(x+6)}^{2}}\;\; &57. \;\;{{(x+1)}^{2}}\;\;\\
58. \;\;{{(x-2)}^{2}}\;\; &59. \;\;{{(2x-3)}^{2}}\;\; &60. \;\;{{(3x-4)}^{2}}\;\;\\
61. \;\;{{(x+4)}^{2}}\;\; &62. \;\;{{(x-7)}^{2}}\;\; &63. \;\;{{(x+8)}^{2}}\;\;\\
64. \;\;{{(x+11)}^{2}}\;\; &65. \;\;{{(5x+3)}^{2}}\;\; &66. \;\;{{(3x+4)}^{2}}\;\;\\
67. \;\;(2x+3)(x-4)\;\; &68. \;\;(3x-1)(x+5)\;\; &69. \;\;{{(x+a)}^{2}}\;\;\\
70. \;\;(3x+2)(2x-5)\;\; &71. \;\;(Kx+t)(Kx-t)\;\; &72. \;\;{{(Kx+t)}^{2}}\;\;\\
73. \;\;(4x+1)(3x-2)\;\; &74. \;\;(5x+3)(x-2)\;\; &75. \;\;{{\left( {{x}^{2}}-3 \right)}^{2}}\;\;\\
76. \;\;{{\left( {{x}^{2}}+1 \right)}^{2}}\;\; &77. \;\;{{\left( {{x}^{2}}-2 \right)}^{2}}\;\; &78. \;\;{{\left( {{x}^{2}}+4x \right)}^{2}}\;\;\\
79. \;\;{{\left( 3{{x}^{2}}+1 \right)}^{2}}\;\; &80. \;\;{{\left( 4{{x}^{2}}-3 \right)}^{2}}\;\; &81. \;\;{{\left( {{x}^{2}}+y \right)}^{2}}\;\;\\
82. \;\;{{\left( 5x-2{{t}^{2}} \right)}^{2}}\;\; &83. \;\;{{\left( 5x-2{{x}^{2}} \right)}^{2}}\;\; &84. \;\;{{\left( 5{{K}^{2}}-2{{t}^{2}} \right)}^{2}}\;\;\\
85. \;\;(2x-1)\left( 3{{x}^{2}}-x-2 \right)\;\; &86. \;\;(x+3)\left( {{x}^{2}}+2x-5 \right)\;\; &87. \;\;(3x-1)\left[ (3x-1)+5K \right]\;\;\\
88. \;\;{{(3x-4K)}^{2}}\;\; &89. \;\;(2t-3K)(2t+3K)\;\; &90. \;\;\left( a+(b-c) \right)\left( a-(b-c) \right)\;\;\\
91. \;\;(x+3)\left( {{x}^{2}}-7x+2 \right)\;\; &92. \;\;(x-2)\left( {{x}^{2}}+4x-3 \right)\;\; &93. \;\;\left[ x+(t-2) \right]\left[ x-(t-2) \right]\;\;\\
94. \;\;(x-4)\left( {{x}^{2}}+3x-1 \right)\;\; &95. \;\;(x+3a)(2x-5a)\;\; &96. \;\;\left( (2x+1)-K \right)\left( (2x+1)+K \right)\;\;\\
97. \;\;(3x+7a)(2x-3a) &98. (x+2y)(3x-4y+5)\;\; &99. \;\;(2x-1)\left( 3{{x}^{2}}-2x+1 \right)\;\;\\
\end{array}