**If \(x + 1\) is a factor of \(x^3 + 3x^2 + kx + 4\), find the value of k.**

### Question

If \(x + 1\) is a factor of \(x^3 + 3x^2 + kx + 4\), find the value of k.### Options

The correct answer is A.

### Explanation:

In this question, we're told that \(x + 1\) is a factor of the cubic polynomial \(x^3 + 3x^2 + kx + 4\). Therefore, we can use the Factor Theorem, which is a special case of the Remainder Theorem. The Factor Theorem states that if a polynomial \(f(x)\) has a factor of the form \(x - a\), then \(f(a) = 0\).

Since \(x + 1\) is a factor of our polynomial, we can rewrite it as \(x - (-1)\). So, in this case, \(a = -1\).

Then we substitute \(x = -1\) into the polynomial, making it equal to 0, because of the Factor Theorem. We get:

\((-1)^3 + 3(-1)^2 + k(-1) + 4 = 0\)Simplifying that equation gives us:

\(-1 + 3 - k + 4 = 0\)This further simplifies to:

\(6 - k = 0\)Finally, solving for k we find that:

\(k = 6\)So, the correct answer is Option A: 6.

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x+1=0

x=0-1

x=-1

Input x=-1 into the equation

x³ + 3x² + kx + 4

(-1)³ + 3(-1)² + k(-1) + 4 = 0

-1 + 3(1) + (-k) + 4 = 0

-1 + 3 -k +4 =0

-1+3+4 = 0+k

K = 6

× + 1 is said to be the factor

So therfore × = -1

Substitute for × where necessary

(-1) 3 + 3 (-1) 2 + k (-1) +4 = 0

-1 + 3 - k + 4 = 0

-1 + 7 = k

K = 6