幾條數學題

1.
　(x – y)(x^4 + x³y + x²y² + xy³ + y^4)
= x(x^4 + x³y + x²y² + xy³ + y^4) – y(x^4 + x³y + x²y² + xy³ + y^4)
= x^5 + x^4 y + x³y² + x²y³ + xy^4 – x^4 y – x³y² – x²y³ – xy^4 – y^5)
= x^5 – y^5

2.
Let x = a + b, y = c + d,
Then (a + b + c + d)(a + b – c – d)
= (a + b + c + d)(a + b – (c + d))
= (x + y)(x – y)
= x² – y²
= (a + b)² – (c + d)²
= (a² + 2ab + b²) – (c² + 2cd + d²)
= a² + b² – c² – d² + 2ab – 2cd

3.
Let (a_n)xⁿ + (a_(n – 1))x^(n – 1) + (a_(n – 2))x^(n – 2) + …… + (a_3)x³ + (a_2)x² + (a_1)x + a_0 be the polynomial, where n is any non-negative integer and n ≧ 3
Then [(a_n)xⁿ + (a_(n – 1))x^(n – 1) + (a_(n – 2))x^(n – 2) + …… + (a_3)x³ + (a_2)x² + (a_1)x + a_0](x – 2)
= [(a_n)x^(n + 1) + (a_(n – 1))xⁿ + (a_(n – 2))x^(n – 1) + …… + (a_3)x^4 + (a_2)x³ + (a_1)x² + (a_0)x] + [– 2(a_n)xⁿ – 2(a_(n – 1))x^(n – 1) – 2(a_(n – 2))x^(n – 2) – ……– 2(a_3)x³ – 2(a_2)x² – 2(a_1)x – 2(a_0)]
= (a_n)x^(n + 1) + [[(a_(n – 1)) – 2(a_n)]xⁿ + [(a_(n – 2)) – 2(a_(n – 1))]x^(n – 1) + …… + [(a_2) – 2(a_3)]x³ + [(a_1) – 2(a_2)]x² + [(a_0) – 2(a_1)]x] – 2(a_0)

∵This polynomial contains the term – 3x²
∴(a_1) – 2(a_2) = – 3

Let a_2 = t
∴(a_1) – 2t = – 3
a_1 = 2t – 3

∴The required polynomial is [(k_n)xⁿ + (k_(n – 1))x^(n – 1) + (k_(n – 2))x^(n – 2) + …… + (k_3)x³] + tx² + (2t – 3)x + u, where n is any non-negative integer and n ≧ 3, t and u, k_n, k_(n – 1), k_(n – 2), ……, k_3 are any real numbers

1. (x - y)(x^4 + x^3 * y + x^2 * y^2 + x * y^3 + y^4)
= x (x^4 + x^3 * y + x^2 * y^2 + x * y^3 + y^4) - y (x^4 + x^3 * y + x^2 * y^2 + x * y^3 + y^4)
= (x^5 + x^4 * y + x^3 * y^2 + x^2 * y^3 + x * y^4) -
(x^4 * y + x^3 * y^2 + x^2 * y^3 + x * y^4 + y^5)
= x^5 + x^4 * y + x^3 * y^2 + x^2 * y^3 + x * y^4 - x^4 * y - x^3 * y^2 - x^2 * y^3 - x * y^4 - y^5
= x^5 - y^5

2. Let x = a + b, y = c + d, then
(a + b + c + d)(a + b - c - d)
= (x + y)(x - y)
= x^2 - y^2
= (a + b) ^ 2 - (c + d)^ 2
= (a^2 + b^2 + 2ab) - (c^2 + d^2 + 2cd)
= a^2 + b^2 - c^2 - d^2 + 2 (ab - cd)

3. Let f(x) be the polynomial, such that f(x) * (x - 2) = - 3x^2 + g(x), for some polynomial g(x) of degree less than 2. By trying (-3x^2) / x = -3x, we can see that (-3x) * (x - 2) = -3x^2 + 6x. So, f(x) = -3x must be one of the answers. By trying (-3x^2) / (-2) = 3/2 * x^2, we can see that (3/2 * x^2) * (x - 2) = 3/2 * x^3 - 3x^2. So f(x) = 3/2 * x^2 will be another answer. Hope that I have not misunderstood your question.

1.x^5-y^5