To find \( g(7) \) for the function \( g(x) = 3x^2 + 4x - 5 \), we can first perform the polynomial long division of \( g(x) \) by \( x - 7 \).
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Set up the division:
We are dividing \( 3x^2 + 4x - 5 \) by \( x - 7 \).
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Perform the division:
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Divide the leading term: \( 3x^2 \div x = 3x \).
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Multiply \( 3x \) by \( x - 7 \): \( 3x(x - 7) = 3x^2 - 21x \).
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Subtract: \[ (3x^2 + 4x - 5) - (3x^2 - 21x) = 4x + 21x - 5 = 25x - 5 \]
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Now, divide \( 25x \) by \( x \): \( 25x \div x = 25 \).
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Multiply \( 25 \) by \( x - 7 \): \( 25(x - 7) = 25x - 175 \).
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Subtract: \[ (25x - 5) - (25x - 175) = -5 + 175 = 170 \]
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Completion of division: After performing the division, we find: \[ g(x) = (x - 7)(3x + 25) + 170 \]
Thus, the quotient is \( 3x + 25 \) and the remainder is \( 170 \). Therefore, we have: \[ g(x) = (x - 7)(3x + 25) + 170 \]
- Evaluate \( g(7) \): From the form of the polynomial we derived, we can see that the value of \( g(7) \) is simply the remainder: \[ g(7) = 170 \]
Hence, \[ \boxed{170} \] is the value of \( g(7) \).