Find a polynomial f(x) of degree 5 such that both of these properties hold:
* f(x) is divisible by x^3
* f(x) + 1 is divisible by (x + 1)(x + 2)(x + 3)
Write your answer in expanded form.
+195 Since f(x) is divisible by x^3, we can write:
f(x) = x^3 g(x)
where g(x) is some polynomial of degree 2. We want f(x) + 1 to be divisible by (x + 1)(x + 2)(x + 3), so we can write:
f(x) + 1 = x^3 g(x) + 1 = (x + 1)(x + 2)(x + 3) h(x)
where h(x) is some polynomial of degree 2.
Expanding both sides of the equation, we get:
x^3 g(x) + 1 = (x + 1)(x + 2)(x + 3) h(x)
x^3 g(x) + 1 = (x^2 + 3x + 2)(x + 3) h(x)
x^3 g(x) + 1 = (x^3 + 3x^2 + 2x + 3x^2 + 9x + 6) h(x)
x^3 g(x) + 1 = x^3 h(x) + 6x^2 h(x) + 11x h(x) + 6h(x)
Comparing coefficients of the same powers of x on both sides of the equation, we get:
g(x) = h(x)
6g(x) = 11h(x)
6h(x) = 1
Solving for h(x), we get:
h(x) = 1/6
Therefore, g(x) = h(x) = 1/6, and we have:
f(x) = x^3 g(x) = x^3/6
Expanding this polynomial, we get:
f(x) = (1/6) x^3
Therefore, the polynomial f(x) = (1/6) x^3 satisfies both properties.
