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Find the remainder when f(x) = 3x^3 – 5x^2 – 16x + 12 is divided by (x-2)

User Ento
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2 Answers

14 votes
14 votes

Final Answer:

The remainder when
\( f(x) = 3x^3 - 5x^2 - 16x + 12 \) \\is divided by
\( f(x) = 3x^3 - 5x^2 - 16x + 12 \)
\( f(x) = 3x^3 - 5x^2 - 16x + 12 \).

Step-by-step explanation:

To find the remainder, we can use the Remainder Theorem, which states that if you divide a polynomial
\( f(x) \) by \( (x - c) \), the remainder is
\( f(c) \). In this case,
\( c = 2 \).

Evaluate
\( f(2) \) by substituting
\( x = 2 \) into the polynomial
\( f(x) = 3x^3 - 5x^2 - 16x + 12 \):\[ f(2) = 3(2)^3 - 5(2)^2 - 16(2) + 12 = 24 - 20 - 32 + 12 = 18 \]

Therefore, the remainder when
\( f(x) \) is divided by
\( (x-2) \) is \( 18 \).

This result makes sense intuitively. When you divide a cubic polynomial by a linear factor, the remainder is a constant term, and in this case, it is
\( 18 \). The division process involves subtracting multiples of
\( (x-2) \) from
\( 3x^3 - 5x^2 - 16x + 12 \) until we can no longer do so, and the remaining constant term is the remainder.

User Curlyreggie
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5 votes
5 votes
(3x + 2) •(x + 3) •(x - 2)
User LarZuK
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2.7k points