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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.

 Mar 7, 2023
 #1
avatar+195 
+1

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.

 Mar 7, 2023
 #2
avatar+33616 
+1

"Therefore, the polynomial f(x) = (1/6) x^3 satisfies both properties."  Note that (1/6)x^3 is not a degree 5 polynomial.

Alan  Mar 7, 2023

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