Question

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

Answer 1

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.

Answer 2

（3x + 2) •（x + 3) •（x - 2）