Question

When a polynomial 2x^3+3x^2+kx+4 leaves a remainder 2(4-k) when divided by (x-2), find the value of k.​

Answer 1

`k = -6`.

Explanation:

Apply long division:

`\begin{aligned}& \;\;\phantom{2\, x^(3) + } 2\, x^(2) + 7\, x + (k + 14)\\ & \, \begin{aligned} x - 2 & \\[-1.7em] & \overline{ \begin{aligned}\smash{)}& 2\, x^(3) + 3\, x^(2) + k\, x + 4 \\[-0.5em] & 2\, x^(3) - 14 \\ &\overline{\phantom{2\, x^(3) + }\begin{aligned} & 7\, x^(2) - k\, x \\[-0.5em] & 7\, x^(2) - 14\, x \\ &  \overline{\begin{aligned} \phantom{7\, x^(2) \phantom{7\, x^(2)}} &(k + 14)\, x + 4 \\[-0.5em] & (k + 14)\, x- 2\, (k + 14) \\ & \overline{\phantom{(k </p><p>+ 14)\, x +}2\, k + 32\quad}\end{aligned}}\end{aligned}}\end{aligned}}\end{aligned}\end{aligned}`

In other words:

`\begin{aligned} & 2\, x^(3) + 3\, x^(2) + k\, x + 4 \\ =\; & (x - 2)\, (2\, x^(2) + 7\, x + (k + 14)) + (2\, k + 32)\end{aligned}`.

The remainder is
`(2\, k + 32)`.

The question states that this remainder may also be expressed as
`2\, (4 - k)`. Equate these two expressions for the remainder and solve for
`k`:

`2\, k + 32 = 2\, (4 - k)`.

`k = -6`.

Substitute
`k = -6` back into expand the expression
`(x - 2)\, (2\, x^(2) + 7\, x + (k + 14)) + (2\, k + 32)`. Expand and verify that the expression indeed matches
`(2\, x^(3) + 3\, x^(2) + k\, x + 4)` with
`k = -6\!`.

`\begin{aligned} & (x - 2)\, (2\, x^(2) + 7\, x + (k + 14)) + (2\, k + 32) \\ =\; &2\, x^(3) + 7\, x^(2) + 8\, x\\ &\quad\quad - 4\, x^(2) -14\, x - 16 + 20 \\ =\; & 2\, x^(3) + 3\, x^(2) - 6\, x + 4 \end{aligned}`.

Answer 2

Remainder Theorem

Answers:

`k = -6`

Explanation:

According to the Remainder Theorem, if we divide a polynomial,
`P(x)`, by
`x -\blue a`, the remainder is
`P(\blue a)`.

We can let
`P(x) = 2x^3 +3x^2 +kx +4`. If we want our remainder to be
`2(4 -k)` when we divide
`P(x)` by
`x -\blue 2`, then
`P(\blue 2) = 2(4 -k)`

Solving for
`k`:

`P(\blue 2) = 2(4 -k) \\ 2(\blue 2)^3 +3(\blue 2)^2 +k(\blue 2) +4 = 2(4 -k) \\ 2(8) +3(4) +2k +4 = 2(4 -k) \\ 16 +12 +2k +4 = 8 -2k \\ 32 +2k = 8 -2k \\ 32 +2k +2k = 8 \\ 32 +4k = 8 \\ 4k = 8 -32 \\ 4k = -24 \\ k = (-24)/(4) \\ k = -6`

The value of
`k` is
`-6`