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# Polynomial

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Let f(x) be the polynomial f(x) = x^7 - 3x^3 + 2.

If g(x) = f(x+7), what is the sum of the coefficients of g(x)?

Jun 15, 2022

#2
+124696
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g (x) =  f (x + 7) =    (x + 7)^7  - 3(x + 7)^3  + 2    =

1x^7 + 49 x^6 + 1029 x^5 + 12005 x^4 + 84032 x^3 + 352884 x^2 + 823102 x + 822516

Jun 15, 2022
#4
+9461
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Alternative solution:

The sum of coefficients of a polynomial $$p(x)$$ is actually just $$p(1)$$.

Proof: (You can omit this part if you just want the answer instead of the explanation.)

Let $$p(x) = a_n x^n + a_{n - 1}x^{n - 1} + a_{n - 2}x^{n - 2} + \cdots + a_2 x^2 + a_1 x + a_0$$.

Then $$p(1) = a_n 1^n + a_{n - 1} 1^{n- 1} + \cdots + a_1 \cdot 1 + a_0 = a_n + a_{n -1} + a_{n-2} + \cdots + a_1 + a_0$$, which is exactly the sum of coefficients of p(x).

Therefore, we just find $$g(1) = f(1 + 7) = f(8)$$. The sum of coefficients of g(x) is $$f(8) = 8^7 - 3\cdot 8^3 + 2$$.

Jun 16, 2022