If $4x = p(x + 3) + q(x - 1)$ is an identity, then the values of $p$ and $q$ are

### Concept & Strategy This question involves a **Polynomial Identity**. An identity means the equation holds true for *any* possible value of $x$. We can solve this using two main strategies: 1. **Coefficient Equating:** Expand the right side and equate the coefficients of $x$ and the constant terms to the left side. 2. **Strategic Substitution:** Choose clever values for $x$ that make terms cancel out (become zero), allowing instant evaluation of $p$ and $q$. (This is the faster method). ### Step-by-Step Solution (Substitution Method) Since the equation is an identity, it works for all values of $x$. We pick values that make the brackets zero. * **Step 1: Solve for p by eliminating q** To make the $q$ term disappear, we set $x - 1 = 0$, which means $x = 1$. Substitute $x = 1$ into the entire equation: $$4(1) = p(1 + 3) + q(1 - 1)$$ $$4 = p(4) + 0$$ $$4 = 4p$$ $$p = 1$$ * **Step 2: Solve for q by eliminating p** To make the $p$ term disappear, we set $x + 3 = 0$, which means $x = -3$. Substitute $x = -3$ into the entire equation: $$4(-3) = p(-3 + 3) + q(-3 - 1)$$ $$-12 = p(0) + q(-4)$$ $$-12 = -4q$$ $$q = \frac{-12}{-4} = 3$$ ### Exam Strategy & Shortcut The Strategic Substitution method shown above *is* the ultimate exam shortcut for identity problems. It bypasses the need to create and solve simultaneous equations entirely. The moment you see $(x-1)$ and $(x+3)$, you should mentally plug in $x=1$ (yielding $4=4p \rightarrow p=1$) and look at the options. Options (a), (b), and (c) all have $p=1$. Then quickly plug in $x=-3$ (yielding $-12=-4q \rightarrow q=3$). The answer is (b). ### Common Pitfall Students often default to expanding the expression: $4x = px + 3p + qx - q$, then grouping: $4x = (p+q)x + (3p-q)$. This creates a system of equations: $p+q=4$ and $3p-q=0$. While this method is mathematically sound, it is time-consuming and introduces more opportunities for basic addition/subtraction errors compared to strategic substitution. ### Final Answer Therefore, the correct answer is **1, 3**.