1. Problem: Multiply the binomials $(5x + 7)$ and $(3x + 4)$.
Formula: Use the distributive property (FOIL method) for binomials:
$$ (a + b)(c + d) = ac + ad + bc + bd $$
Step 1: Multiply the first terms: $5x \times 3x = 15x^2$.
Step 2: Multiply the outer terms: $5x \times 4 = 20x$.
Step 3: Multiply the inner terms: $7 \times 3x = 21x$.
Step 4: Multiply the last terms: $7 \times 4 = 28$.
Step 5: Add all terms: $15x^2 + 20x + 21x + 28$.
Step 6: Combine like terms: $15x^2 + 41x + 28$.
Answer: $$15x^2 + 41x + 28$$
2. Problem: Multiply $(4x + 9)$ and $(x - 6)$.
Step 1: $4x \times x = 4x^2$.
Step 2: $4x \times (-6) = -24x$.
Step 3: $9 \times x = 9x$.
Step 4: $9 \times (-6) = -54$.
Step 5: Sum: $4x^2 - 24x + 9x - 54 = 4x^2 - 15x - 54$.
Answer: $$4x^2 - 15x - 54$$
3. Problem: Multiply $(2x + 5)$ and $(4x - 3)$.
Step 1: $2x \times 4x = 8x^2$.
Step 2: $2x \times (-3) = -6x$.
Step 3: $5 \times 4x = 20x$.
Step 4: $5 \times (-3) = -15$.
Step 5: Sum: $8x^2 - 6x + 20x - 15 = 8x^2 + 14x - 15$.
Answer: $$8x^2 + 14x - 15$$
4. Problem: Multiply $(-8)$ and $(5y - 1)$.
Step 1: $-8 \times 5y = -40y$.
Step 2: $-8 \times (-1) = 8$.
Step 3: Sum: $-40y + 8$.
Answer: $$-40y + 8$$
5. Problem: Multiply $(7x + 2y)$ and $(x + 4y)$.
Step 1: $7x \times x = 7x^2$.
Step 2: $7x \times 4y = 28xy$.
Step 3: $2y \times x = 2xy$.
Step 4: $2y \times 4y = 8y^2$.
Step 5: Sum: $7x^2 + 28xy + 2xy + 8y^2 = 7x^2 + 30xy + 8y^2$.
Answer: $$7x^2 + 30xy + 8y^2$$
6. Problem: Multiply $(9x + 5y)$ and $(4x + 3y)$.
Step 1: $9x \times 4x = 36x^2$.
Step 2: $9x \times 3y = 27xy$.
Step 3: $5y \times 4x = 20xy$.
Step 4: $5y \times 3y = 15y^2$.
Step 5: Sum: $36x^2 + 27xy + 20xy + 15y^2 = 36x^2 + 47xy + 15y^2$.
Answer: $$36x^2 + 47xy + 15y^2$$
7. Problem: Multiply $(3m - 4n)$ and $(2m - 3n)$.
Step 1: $3m \times 2m = 6m^2$.
Step 2: $3m \times (-3n) = -9mn$.
Step 3: $-4n \times 2m = -8mn$.
Step 4: $-4n \times (-3n) = 12n^2$.
Step 5: Sum: $6m^2 - 9mn - 8mn + 12n^2 = 6m^2 - 17mn + 12n^2$.
Answer: $$6m^2 - 17mn + 12n^2$$
8. Problem: Multiply $(x^2 - a^2)$ and $(x - a)$.
Step 1: $x^2 \times x = x^3$.
Step 2: $x^2 \times (-a) = -a x^2$.
Step 3: $-a^2 \times x = -a^2 x$.
Step 4: $-a^2 \times (-a) = a^3$.
Step 5: Sum: $x^3 - a x^2 - a^2 x + a^3$.
Answer: $$x^3 - a x^2 - a^2 x + a^3$$
9. Problem: Multiply $(x^2 - y^2)$ and $(x + 2y)$.
Step 1: $x^2 \times x = x^3$.
Step 2: $x^2 \times 2y = 2x^2 y$.
Step 3: $-y^2 \times x = -x y^2$.
Step 4: $-y^2 \times 2y = -2 y^3$.
Step 5: Sum: $x^3 + 2x^2 y - x y^2 - 2 y^3$.
Answer: $$x^3 + 2x^2 y - x y^2 - 2 y^3$$
10. Problem: Multiply $(3p^2 + q^2)$ and $(2p^2 - 3q^2)$.
Step 1: $3p^2 \times 2p^2 = 6p^4$.
Step 2: $3p^2 \times (-3q^2) = -9p^2 q^2$.
Step 3: $q^2 \times 2p^2 = 2p^2 q^2$.
Step 4: $q^2 \times (-3q^2) = -3q^4$.
Step 5: Sum: $6p^4 - 9p^2 q^2 + 2p^2 q^2 - 3q^4 = 6p^4 - 7p^2 q^2 - 3q^4$.
Answer: $$6p^4 - 7p^2 q^2 - 3q^4$$
11. Problem: Multiply $(2x^2 - 5y^2)$ and $(x^2 + 3y^2)$.
Step 1: $2x^2 \times x^2 = 2x^4$.
Step 2: $2x^2 \times 3y^2 = 6x^2 y^2$.
Step 3: $-5y^2 \times x^2 = -5x^2 y^2$.
Step 4: $-5y^2 \times 3y^2 = -15 y^4$.
Step 5: Sum: $2x^4 + 6x^2 y^2 - 5x^2 y^2 - 15 y^4 = 2x^4 + x^2 y^2 - 15 y^4$.
Answer: $$2x^4 + x^2 y^2 - 15 y^4$$
12. Problem: Multiply $(x^3 - y^3)$ and $(x^2 + y^2)$.
Step 1: $x^3 \times x^2 = x^5$.
Step 2: $x^3 \times y^2 = x^3 y^2$.
Step 3: $-y^3 \times x^2 = -x^2 y^3$.
Step 4: $-y^3 \times y^2 = -y^5$.
Step 5: Sum: $x^5 + x^3 y^2 - x^2 y^3 - y^5$.
Answer: $$x^5 + x^3 y^2 - x^2 y^3 - y^5$$
13. Problem: Multiply $(x^4 + y^4)$ and $(x^2 - y^2)$.
Step 1: $x^4 \times x^2 = x^6$.
Step 2: $x^4 \times (-y^2) = -x^4 y^2$.
Step 3: $y^4 \times x^2 = x^2 y^4$.
Step 4: $y^4 \times (-y^2) = -y^6$.
Step 5: Sum: $x^6 - x^4 y^2 + x^2 y^4 - y^6$.
Answer: $$x^6 - x^4 y^2 + x^2 y^4 - y^6$$
14. Problem: Multiply $(x^4 + 1/x^4)$ and $(x + 1/x)$.
Step 1: $x^4 \times x = x^5$.
Step 2: $x^4 \times 1/x = x^3$.
Step 3: $1/x^4 \times x = x^{-3}$.
Step 4: $1/x^4 \times 1/x = x^{-5}$.
Step 5: Sum: $x^5 + x^3 + x^{-3} + x^{-5}$.
Answer: $$x^5 + x^3 + x^{-3} + x^{-5}$$
15. Problem: Multiply $(x^2 - 3x + 7)$ and $(2x + 3)$.
Step 1: $x^2 \times 2x = 2x^3$.
Step 2: $x^2 \times 3 = 3x^2$.
Step 3: $-3x \times 2x = -6x^2$.
Step 4: $-3x \times 3 = -9x$.
Step 5: $7 \times 2x = 14x$.
Step 6: $7 \times 3 = 21$.
Step 7: Sum: $2x^3 + 3x^2 - 6x^2 - 9x + 14x + 21 = 2x^3 - 3x^2 + 5x + 21$.
Answer: $$2x^3 - 3x^2 + 5x + 21$$
16. Problem: Multiply $(3x^2 + 5x - 9)$ and $(3x - 5)$.
Step 1: $3x^2 \times 3x = 9x^3$.
Step 2: $3x^2 \times (-5) = -15x^2$.
Step 3: $5x \times 3x = 15x^2$.
Step 4: $5x \times (-5) = -25x$.
Step 5: $-9 \times 3x = -27x$.
Step 6: $-9 \times (-5) = 45$.
Step 7: Sum: $9x^3 - 15x^2 + 15x^2 - 25x - 27x + 45 = 9x^3 - 52x + 45$.
Answer: $$9x^3 - 52x + 45$$
17. Problem: Multiply $(x^2 - xy + y^2)$ and $(x + y)$.
Step 1: $x^2 \times x = x^3$.
Step 2: $x^2 \times y = x^2 y$.
Step 3: $-xy \times x = -x^2 y$.
Step 4: $-xy \times y = -x y^2$.
Step 5: $y^2 \times x = x y^2$.
Step 6: $y^2 \times y = y^3$.
Step 7: Sum: $x^3 + x^2 y - x^2 y - x y^2 + x y^2 + y^3 = x^3 + y^3$.
Answer: $$x^3 + y^3$$
18. Problem: Multiply $(x^2 + xy + y^2)$ and $(x - y)$.
Step 1: $x^2 \times x = x^3$.
Step 2: $x^2 \times (-y) = -x^2 y$.
Step 3: $xy \times x = x^2 y$.
Step 4: $xy \times (-y) = -x y^2$.
Step 5: $y^2 \times x = x y^2$.
Step 6: $y^2 \times (-y) = -y^3$.
Step 7: Sum: $x^3 - x^2 y + x^2 y - x y^2 + x y^2 - y^3 = x^3 - y^3$.
Answer: $$x^3 - y^3$$
19. Problem: Multiply $(x^3 - 2x^2 + 5)$ and $(4x - 1)$.
Step 1: $x^3 \times 4x = 4x^4$.
Step 2: $x^3 \times (-1) = -x^3$.
Step 3: $-2x^2 \times 4x = -8x^3$.
Step 4: $-2x^2 \times (-1) = 2x^2$.
Step 5: $5 \times 4x = 20x$.
Step 6: $5 \times (-1) = -5$.
Step 7: Sum: $4x^4 - x^3 - 8x^3 + 2x^2 + 20x - 5 = 4x^4 - 9x^3 + 2x^2 + 20x - 5$.
Answer: $$4x^4 - 9x^3 + 2x^2 + 20x - 5$$
20. Problem: Multiply $(9x^2 - x + 15)$ and $(x^2 - 3)$.
Step 1: $9x^2 \times x^2 = 9x^4$.
Step 2: $9x^2 \times (-3) = -27x^2$.
Step 3: $-x \times x^2 = -x^3$.
Step 4: $-x \times (-3) = 3x$.
Step 5: $15 \times x^2 = 15x^2$.
Step 6: $15 \times (-3) = -45$.
Step 7: Sum: $9x^4 - 27x^2 - x^3 + 3x + 15x^2 - 45 = 9x^4 - x^3 - 12x^2 + 3x - 45$.
Answer: $$9x^4 - x^3 - 12x^2 + 3x - 45$$
21. Problem: Multiply $(x^2 - 5x + 8)$ and $(x^2 + 2)$.
Step 1: $x^2 \times x^2 = x^4$.
Step 2: $x^2 \times 2 = 2x^2$.
Step 3: $-5x \times x^2 = -5x^3$.
Step 4: $-5x \times 2 = -10x$.
Step 5: $8 \times x^2 = 8x^2$.
Step 6: $8 \times 2 = 16$.
Step 7: Sum: $x^4 + 2x^2 - 5x^3 - 10x + 8x^2 + 16 = x^4 - 5x^3 + 10x^2 - 10x + 16$.
Answer: $$x^4 - 5x^3 + 10x^2 - 10x + 16$$
22. Problem: Multiply $(x^3 - 5x^2 + 3x + 1)$ and $(x^2 - 3)$.
Step 1: $x^3 \times x^2 = x^5$.
Step 2: $x^3 \times (-3) = -3x^3$.
Step 3: $-5x^2 \times x^2 = -5x^4$.
Step 4: $-5x^2 \times (-3) = 15x^2$.
Step 5: $3x \times x^2 = 3x^3$.
Step 6: $3x \times (-3) = -9x$.
Step 7: $1 \times x^2 = x^2$.
Step 8: $1 \times (-3) = -3$.
Step 9: Sum: $x^5 - 5x^4 - 3x^3 + 3x^3 + 15x^2 + x^2 - 9x - 3 = x^5 - 5x^4 + 16x^2 - 9x - 3$.
Answer: $$x^5 - 5x^4 + 16x^2 - 9x - 3$$
23. Problem: Multiply $(3x + 2y - 4)$ and $(x - y + 2)$.
Step 1: $3x \times x = 3x^2$.
Step 2: $3x \times (-y) = -3xy$.
Step 3: $3x \times 2 = 6x$.
Step 4: $2y \times x = 2xy$.
Step 5: $2y \times (-y) = -2y^2$.
Step 6: $2y \times 2 = 4y$.
Step 7: $-4 \times x = -4x$.
Step 8: $-4 \times (-y) = 4y$.
Step 9: $-4 \times 2 = -8$.
Step 10: Sum: $3x^2 - 3xy + 6x + 2xy - 2y^2 + 4y - 4x + 4y - 8 = 3x^2 - xy + 2x - 2y^2 + 8y - 8$.
Answer: $$3x^2 - xy + 2x - 2y^2 + 8y - 8$$
24. Problem: Multiply $(x^2 - 5x + 8)$ and $(x^2 + 2x - 3)$.
Step 1: $x^2 \times x^2 = x^4$.
Step 2: $x^2 \times 2x = 2x^3$.
Step 3: $x^2 \times (-3) = -3x^2$.
Step 4: $-5x \times x^2 = -5x^3$.
Step 5: $-5x \times 2x = -10x^2$.
Step 6: $-5x \times (-3) = 15x$.
Step 7: $8 \times x^2 = 8x^2$.
Step 8: $8 \times 2x = 16x$.
Step 9: $8 \times (-3) = -24$.
Step 10: Sum: $x^4 + 2x^3 - 3x^2 - 5x^3 - 10x^2 + 15x + 8x^2 + 16x - 24 = x^4 - 3x^3 - 5x^2 + 31x - 24$.
Answer: $$x^4 - 3x^3 - 5x^2 + 31x - 24$$
25. Problem: Multiply $(2x^2 + 3x - 7)$ and $(3x^2 - 5x + 4)$.
Step 1: $2x^2 \times 3x^2 = 6x^4$.
Step 2: $2x^2 \times (-5x) = -10x^3$.
Step 3: $2x^2 \times 4 = 8x^2$.
Step 4: $3x \times 3x^2 = 3x^3$.
Step 5: $3x \times (-5x) = -15x^2$.
Step 6: $3x \times 4 = 12x$.
Step 7: $-7 \times 3x^2 = -21x^2$.
Step 8: $-7 \times (-5x) = 35x$.
Step 9: $-7 \times 4 = -28$.
Step 10: Sum: $6x^4 - 10x^3 + 8x^2 + 3x^3 - 15x^2 + 12x - 21x^2 + 35x - 28 = 6x^4 - 7x^3 - 28x^2 + 47x - 28$.
Answer: $$6x^4 - 7x^3 - 28x^2 + 47x - 28$$
26. Problem: Multiply $(9x^2 - x + 15)$ and $(x^2 - x - 1)$.
Step 1: $9x^2 \times x^2 = 9x^4$.
Step 2: $9x^2 \times (-x) = -9x^3$.
Step 3: $9x^2 \times (-1) = -9x^2$.
Step 4: $-x \times x^2 = -x^3$.
Step 5: $-x \times (-x) = x^2$.
Step 6: $-x \times (-1) = x$.
Step 7: $15 \times x^2 = 15x^2$.
Step 8: $15 \times (-x) = -15x$.
Step 9: $15 \times (-1) = -15$.
Step 10: Sum: $9x^4 - 9x^3 - 9x^2 - x^3 + x^2 + x + 15x^2 - 15x - 15 = 9x^4 - 10x^3 + 7x^2 - 14x - 15$.
Answer: $$9x^4 - 10x^3 + 7x^2 - 14x - 15$$
27. Problem: Square $(x + 6)$.
Formula: $(a + b)^2 = a^2 + 2ab + b^2$.
Step 1: $x^2 + 2 \times x \times 6 + 6^2 = x^2 + 12x + 36$.
Answer: $$x^2 + 12x + 36$$
28. Problem: Square $(4x + 5y)$.
Step 1: $(4x)^2 + 2 \times 4x \times 5y + (5y)^2 = 16x^2 + 40xy + 25y^2$.
Answer: $$16x^2 + 40xy + 25y^2$$
29. Problem: Square $(7a + 9b)$.
Step 1: $49a^2 + 126ab + 81b^2$.
Answer: $$49a^2 + 126ab + 81b^2$$
30. Problem: Square $(\frac{2}{3}x + \frac{4}{5}y)$.
Step 1: $(\frac{2}{3}x)^2 + 2 \times \frac{2}{3}x \times \frac{4}{5}y + (\frac{4}{5}y)^2 = \frac{4}{9}x^2 + \frac{16}{15}xy + \frac{16}{25}y^2$.
Answer: $$\frac{4}{9}x^2 + \frac{16}{15}xy + \frac{16}{25}y^2$$
31. Problem: Square $(x^2 + 7)$.
Step 1: $x^4 + 2 \times x^2 \times 7 + 49 = x^4 + 14x^2 + 49$.
Answer: $$x^4 + 14x^2 + 49$$
32. Problem: Square $(\frac{5}{6}a^2 + 2)$.
Step 1: $(\frac{5}{6}a^2)^2 + 2 \times \frac{5}{6}a^2 \times 2 + 4 = \frac{25}{36}a^4 + \frac{10}{3}a^2 + 4$.
Answer: $$\frac{25}{36}a^4 + \frac{10}{3}a^2 + 4$$
33. Problem: Square $(x - 4)$.
Formula: $(a - b)^2 = a^2 - 2ab + b^2$.
Step 1: $x^2 - 2 \times x \times 4 + 16 = x^2 - 8x + 16$.
Answer: $$x^2 - 8x + 16$$
34. Problem: Square $(2x - 3y)$.
Step 1: $4x^2 - 12xy + 9y^2$.
Answer: $$4x^2 - 12xy + 9y^2$$
35. Problem: Square $(\frac{3}{4}x - \frac{5}{6}y)$.
Step 1: $(\frac{3}{4}x)^2 - 2 \times \frac{3}{4}x \times \frac{5}{6}y + (\frac{5}{6}y)^2 = \frac{9}{16}x^2 - \frac{5}{8}xy + \frac{25}{36}y^2$.
Answer: $$\frac{9}{16}x^2 - \frac{5}{8}xy + \frac{25}{36}y^2$$
36. Problem: Square $(x - \frac{3}{x})$.
Step 1: $x^2 - 2 \times x \times \frac{3}{x} + \frac{9}{x^2} = x^2 - 6 + \frac{9}{x^2}$.
Answer: $$x^2 - 6 + \frac{9}{x^2}$$
37. Problem: Square $(\frac{1}{3}x^2 - 9)$.
Step 1: $(\frac{1}{3}x^2)^2 - 2 \times \frac{1}{3}x^2 \times 9 + 81 = \frac{1}{9}x^4 - 6x^2 + 81$.
Answer: $$\frac{1}{9}x^4 - 6x^2 + 81$$
38. Problem: Square $(\frac{1}{2}y^2 - \frac{1}{3}y)$.
Step 1: $(\frac{1}{2}y^2)^2 - 2 \times \frac{1}{2}y^2 \times \frac{1}{3}y + (\frac{1}{3}y)^2 = \frac{1}{4}y^4 - \frac{1}{3}y^3 + \frac{1}{9}y^2$.
Answer: $$\frac{1}{4}y^4 - \frac{1}{3}y^3 + \frac{1}{9}y^2$$
39. Problem: Expand $(8a + 3b)^2$.
Step 1: $64a^2 + 48ab + 9b^2$.
Answer: $$64a^2 + 48ab + 9b^2$$
40. Problem: Expand $(7x + 2y)^2$.
Step 1: $49x^2 + 28xy + 4y^2$.
Answer: $$49x^2 + 28xy + 4y^2$$
41. Problem: Expand $(5x + 11)^2$.
Step 1: $25x^2 + 110x + 121$.
Answer: $$25x^2 + 110x + 121$$
42. Problem: Expand $(\frac{a}{2} + \frac{2}{a})^2$.
Step 1: $(\frac{a}{2})^2 + 2 \times \frac{a}{2} \times \frac{2}{a} + (\frac{2}{a})^2 = \frac{a^2}{4} + 2 + \frac{4}{a^2}$.
Answer: $$\frac{a^2}{4} + 2 + \frac{4}{a^2}$$
43. Problem: Expand $(\frac{3x}{4} + \frac{2y}{9})^2$.
Step 1: $(\frac{3x}{4})^2 + 2 \times \frac{3x}{4} \times \frac{2y}{9} + (\frac{2y}{9})^2 = \frac{9x^2}{16} + \frac{xy}{3} + \frac{4y^2}{81}$.
Answer: $$\frac{9x^2}{16} + \frac{xy}{3} + \frac{4y^2}{81}$$
44. Problem: Expand $(9x - 10)^2$.
Step 1: $81x^2 - 180x + 100$.
Answer: $$81x^2 - 180x + 100$$
45. Problem: Expand $(x^2 y - y z^2)^2$.
Step 1: $(x^2 y)^2 - 2 \times x^2 y \times y z^2 + (y z^2)^2 = x^4 y^2 - 2 x^2 y^2 z^2 + y^2 z^4$.
Answer: $$x^4 y^2 - 2 x^2 y^2 z^2 + y^2 z^4$$
46. Problem: Expand $(\frac{x}{y} - \frac{y}{x})^2$.
Step 1: $(\frac{x}{y})^2 - 2 \times \frac{x}{y} \times \frac{y}{x} + (\frac{y}{x})^2 = \frac{x^2}{y^2} - 2 + \frac{y^2}{x^2}$.
Answer: $$\frac{x^2}{y^2} - 2 + \frac{y^2}{x^2}$$
47. Problem: Expand $(3m - \frac{4}{5}n)^2$.
Step 1: $9m^2 - 2 \times 3m \times \frac{4}{5}n + \frac{16}{25}n^2 = 9m^2 - \frac{24}{5}mn + \frac{16}{25}n^2$.
Answer: $$9m^2 - \frac{24}{5}mn + \frac{16}{25}n^2$$
48. Problem: Multiply $(x + 3)(x - 3)$.
Formula: Difference of squares: $(a + b)(a - b) = a^2 - b^2$.
Step 1: $x^2 - 9$.
Answer: $$x^2 - 9$$
49. Problem: Multiply $(2x + 5)(2x - 5)$.
Step 1: $4x^2 - 25$.
Answer: $$4x^2 - 25$$
50. Problem: Multiply $(8 + x)(8 - x)$.
Step 1: $64 - x^2$.
Answer: $$64 - x^2$$
51. Problem: Multiply $(7x + 11y)(7x - 11y)$.
Step 1: $49x^2 - 121y^2$.
Answer: $$49x^2 - 121y^2$$
52. Problem: Multiply $(5x^2 + \frac{3}{4} y^2)(5x^2 - \frac{3}{4} y^2)$.
Step 1: $25x^4 - \frac{9}{16} y^4$.
Answer: $$25x^4 - \frac{9}{16} y^4$$
53. Problem: Multiply $(\frac{4x}{5} - \frac{5y}{3})(\frac{4x}{5} + \frac{5y}{3})$.
Step 1: $\frac{16x^2}{25} - \frac{25y^2}{9}$.
Answer: $$\frac{16x^2}{25} - \frac{25y^2}{9}$$
54. Problem: Multiply $(x + \frac{1}{x})(x - \frac{1}{x})$.
Step 1: $x^2 - \frac{1}{x^2}$.
Answer: $$x^2 - \frac{1}{x^2}$$
55. Problem: Multiply $(\frac{1}{x} + \frac{1}{y})(\frac{1}{x} - \frac{1}{y})$.
Step 1: $\frac{1}{x^2} - \frac{1}{y^2}$.
Answer: $$\frac{1}{x^2} - \frac{1}{y^2}$$
56. Problem: Multiply $(2a + \frac{3}{b})(2a - \frac{3}{b})$.
Step 1: $4a^2 - \frac{9}{b^2}$.
Answer: $$4a^2 - \frac{9}{b^2}$$
Polynomial Products
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