» » » If $$x + 1$$ is a factor of $$x^3 + 3x^2 + kx + 4$$, find the value of k.

# 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.

A)
6
B)
4
C)
-4
D)
3

### 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.

## Discussion (2)

• 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