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If f(x) = x^4 + ax^3 + bx^2 + cx + d, and f(2)= 1, f(3) = 2, f(4) = 3, f(5) = 4, what is the value of a + b + c + d ? Please find it out as soon as possible. I am having problems in finding it out

Note by Eshan Abbas
2 years, 7 months ago

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Let \( g(x) = f(x) - x + 1 \).Then \(g(x) \) is of degree \( 4\). And for \( x =2,3,4,5 \), \( g(x) = 0 \).This implies \( 2,3,4,5 \) are the roots of \( g(x) \).

Thus we can write \( g(x) = (x-2)(x-3)(x-4)(x-5) \)

\( \Rightarrow f(x) - x + 1 = (x-2)(x-3)(x-4)(x-5) \)

\( \Rightarrow x^4 + ax^3 + bx^2 + cx + d - x + 1 = (x-2)(x-3)(x-4)(x-5) \).

Putting \( x = 1 \)

\( \Rightarrow 1^4 + a*1^3 + b*1^2 + c*1 + d - 1 + 1 = (1-2)(1-3)(1-4)(1-5) \)

\( \Rightarrow a+b+c+d = 24-1 = 23 \). Siddhartha Srivastava · 2 years, 7 months ago

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@Siddhartha Srivastava I think that g(x) should be equal to f(x) - x + 1 Eshan Abbas · 2 years, 7 months ago

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@Eshan Abbas Thanks. Corrected. Hopefully correct now. Siddhartha Srivastava · 2 years, 7 months ago

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@Eshan Abbas thats what he got bro Saurav Zuer · 2 years, 6 months ago

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