Pregunta
Given \( g(x)=-3 x^{5}+7 x^{4}+9 x^{2}-x+9 \), Part 1 of 2 (a) Evaluate \( g(2) \). \( g(2)= \) Part 2 of 2 (b) Determine the remainder when \( g(x) \) is divided by \( (x-2) \). The remainder is
Ask by Bob Chan. in the United States
Mar 15,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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\( g(2) = 59 \) and the remainder when \( g(x) \) is divided by \( (x-2) \) is 59.
Solución
**(a)** To evaluate \( g(2) \), substitute \( x = 2 \) into \( g(x) \):
\[
g(2) = -3(2)^5 + 7(2)^4 + 9(2)^2 - (2) + 9
\]
Step-by-step calculations:
1. \( -3(2)^5 = -3 \times 32 = -96 \)
2. \( 7(2)^4 = 7 \times 16 = 112 \)
3. \( 9(2)^2 = 9 \times 4 = 36 \)
4. \( - (2) = -2 \)
5. Constant term \( = 9 \)
Now, sum all the terms:
\[
-96 + 112 + 36 - 2 + 9 = 59
\]
Thus,
\[
g(2) = 59.
\]
**(b)** By the Remainder Theorem, the remainder when \( g(x) \) is divided by \( (x-2) \) is \( g(2) \). Hence, the remainder is:
\[
59.
\]
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Simplifique esta solución Beyond the Answer
To evaluate \( g(2) \): First, substitute \( x = 2 \) into the function: \[ g(2) = -3(2)^5 + 7(2)^4 + 9(2)^2 - (2) + 9 \] Calculating each term: \[ = -3(32) + 7(16) + 9(4) - 2 + 9 \] \[ = -96 + 112 + 36 - 2 + 9 \] \[ = -96 + 112 = 16 \] \[ 16 + 36 = 52 \] \[ 52 - 2 = 50 \] \[ 50 + 9 = 59 \] So, \( g(2) = 59 \). Now, for the remainder when \( g(x) \) is divided by \( (x-2) \), we can use the Remainder Theorem. According to the theorem, the remainder of \( g(x) \) when divided by \( (x - a) \) is simply \( g(a) \). Since we already calculated \( g(2) \): The remainder is \( 59 \).
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