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1. Let f(x) be a polynomial such that f(0)=4, f(1)=5, and f(2)=10. Find the remainder when f(x) is divided by x(x-1)(x-2).

2. Find a polynomial f(x) of degree 5 such that both of these properties hold:

f(x) is divisible by \(x^3\)

f(x)+2 is divisible by \((x+1)^3\)

 May 31, 2020
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1. The remainder works out to 2x^2 - 4x + 3.

 

2. Since f(x) is divisible by x^3, f(x) is of the form ax^5 + bx^4 + cx^3.

 

You then want ax^5 + bx^4 + cx^3 + 2 to be divisible by (x + 1)^3.  Using long division, you get the equations

-10a  + 6b - 3c = 0

4a - 3b + 2c = 0

-a + b - c + 2 = 0

==> a = 6, b = 16, c = 12

 

So f(x) = 6x^5 + 16x^4 + 12x^3.

 May 31, 2020

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