If (x + 2) and (x - 1) are factors of the expression $Lx + 2kx^{2} + 24$, find the values of L and k.

1. Analyze the Question

We are given a polynomial $P(x) = 2kx^2 + Lx + 24$ (rearranging the terms for clarity). We are told that $(x + 2)$ and $(x - 1)$ are factors of this expression. According to the Factor Theorem, if $(x - a)$ is a factor of $P(x)$, then $P(a) = 0$.

2. Explain the Concept

• Factor Theorem: If $(x + 2)$ is a factor, then $P(-2) = 0$.
• Factor Theorem: If $(x - 1)$ is a factor, then $P(1) = 0$. We will use these two conditions to create a system of simultaneous equations to solve for $L$ and $k$.

3. Show the Solution

Given $P(x) = 2kx^2 + Lx + 24$:

Step 1: Use the factor $(x + 2)$ Set $x = -2$: $P(-2) = 2k(-2)^2 + L(-2) + 24 = 0$ $2k(4) - 2L + 24 = 0$ $8k - 2L + 24 = 0$ Divide by 2 to simplify: $4k - L + 12 = 0$ $4k - L = -12$ --- (Equation 1)

Step 2: Use the factor $(x - 1)$ Set $x = 1$: $P(1) = 2k(1)^2 + L(1) + 24 = 0$ $2k + L + 24 = 0$ $2k + L = -24$ --- (Equation 2)

Step 3: Solve the simultaneous equations Add Equation 1 and Equation 2: $(4k - L) + (2k + L) = -12 + (-24)$ $6k = -36$ $k = -36 / 6$ $k = -6$

Substitute $k = -6$ into Equation 2: $2(-6) + L = -24$ $-12 + L = -24$ $L = -24 + 12$ $L = -12$

4. Identify the Answer

Comparing our results ($L = -12, k = -6$) with the given options, we find they match Option A.

5. Tip

When you see "factors" in a polynomial question, always think "Factor Theorem." Immediately set the expression to zero using the root of the factor (e.g., for $x+2$, use $-2$). If you have two variables to find, you will always need two factors to create two equations.