If X = -2, then X^3 - X^2 - X - 1 is equal to

1. Rules of Signs for Integers (basic)

Welcome to your first step in mastering quantitative aptitude! To navigate complex mathematical problems, we must first master the Rules of Signs. At its simplest, a sign (+ or -) indicates a direction or a relative value. For example, in geography, we use signs to denote time zones: a positive sign indicates hours added to Greenwich Mean Time (GMT), while a negative sign indicates hours subtracted from it Exploring Society: India and Beyond, Locating Places on the Earth, p.21. Understanding how these signs interact is the bedrock of algebra and arithmetic.

When adding or subtracting integers, the sign of the result depends on the "strength" (magnitude) of the numbers. If you add two negative numbers, the result becomes more negative (e.g., -5 + -3 = -8). If you add numbers with different signs, you find the difference between them and keep the sign of the number with the larger absolute value. A vital rule to remember is that subtracting a negative is equivalent to addition. For instance, -8 - (-2) simplifies to -8 + 2, which equals -6.

For multiplication and division, the rules follow a very consistent pattern based on whether the signs are the same or different. These rules are non-negotiable and apply regardless of how large the numbers are:

Operation	Sign of Integer A	Sign of Integer B	Resulting Sign
Multiplication/Division	Positive (+)	Positive (+)	Positive (+)
Multiplication/Division	Negative (-)	Negative (-)	Positive (+)
Multiplication/Division	Positive (+)	Negative (-)	Negative (-)
Multiplication/Division	Negative (-)	Positive (+)	Negative (-)

When dealing with exponents, an even power of a negative number always results in a positive value (e.g., (-2)² = 4), while an odd power of a negative number remains negative (e.g., (-2)³ = -8). Mastering these small shifts in signs will prevent the most common "silly mistakes" in competitive exams.

Sources: Exploring Society: India and Beyond, Locating Places on the Earth, p.21

2. Exponents and Powers with Negative Bases (basic)

In our journey through quantitative aptitude, understanding how exponents interact with negative bases is a fundamental building block. An exponent tells us how many times to multiply a base by itself. However, when the base is negative, the outcome depends entirely on whether the exponent is an even or odd number. Just as electricity requires identifying a negative terminal to function correctly Science-Class VII NCERT (Revised ed 2025), Electricity: Circuits and their Components, p.25, in mathematics, we must identify the sign of our base before performing any operations.

The golden rules for negative bases are:

• Even Exponents: If a negative number is raised to an even power (2, 4, 6...), the result is always positive. For example, (-2)² = (-2) × (-2) = 4.
• Odd Exponents: If a negative number is raised to an odd power (1, 3, 5...), the result remains negative. For example, (-2)³ = (-2) × (-2) × (-2) = -8.

Think of it as pairs: every pair of negative signs cancels out to become positive. If you have an odd number, one negative sign is always left without a partner, keeping the final product negative.

A common trap in competitive exams is the Parentheses Trap. There is a massive difference between (-3)² and -3². In (-3)², the negative sign is part of the base being squared, resulting in 9. In -3², the exponent only applies to the 3, and the negative sign is applied after the squaring, resulting in -9. Much like how salts are grouped into families based on their negative radicals Science, Class X (NCERT 2025 ed.), Acids, Bases and Salts, p.29, we must group our signs and numbers carefully using brackets to ensure the exponent acts on the correct 'family' of terms.

Sources: Science-Class VII NCERT (Revised ed 2025), Electricity: Circuits and their Components, p.25; Science, Class X (NCERT 2025 ed.), Acids, Bases and Salts, p.29

3. Structure of Algebraic Expressions (basic)

Algebraic expressions are the fundamental building blocks of quantitative logic. At their core, they consist of variables (letters like x or y representing unknown quantities) and constants (fixed numerical values). These components are combined using operators such as addition, subtraction, and multiplication. Each part of an expression separated by a plus or minus sign is called a term. For example, in the expression x³ - x² - x - 1, there are four distinct terms.

To master these structures, we must identify the coefficient—the numerical factor multiplying a variable. Just as the "ranking coefficient method" is used to simplify complex agricultural data into a single comparable value Geography of India, Spatial Organisation of Agriculture, p.10, identifying coefficients allows us to understand the weight of each term. Furthermore, we categorize expressions by their degree, which is the highest exponent of the variable. A "cubic" expression has a degree of 3, meaning the highest power is x³.

The most vital skill is evaluation by substitution. This is the process of replacing the variable with a specific number to find the expression's total value. When substituting, we must be exceptionally careful with signs and exponents:

• Even Powers: A negative number raised to an even power becomes positive (e.g., (-2)² = 4).
• Odd Powers: A negative number raised to an odd power remains negative (e.g., (-2)³ = -8).

Just as specific exclusions in a definition (like the exclusion of certain water bodies from the wetland category) can fundamentally change the "status" of a result Environment, Aquatic Ecosystem, p.44, failing to correctly apply the negative sign during substitution will change the final value of your polynomial. For instance, evaluating x³ - x² - x - 1 for x = -2 follows this logical flow: (-8) - (4) - (-2) - 1. Simplifying stepwise, we get -12 + 2 - 1, resulting in a final value of -11.

Sources: Geography of India, Spatial Organisation of Agriculture, p.10; Environment, Aquatic Ecosystem, p.44

4. The VBODMAS Rule for Simplification (intermediate)

In competitive exams like the UPSC CSAT, the VBODMAS rule is the fundamental protocol that ensures there is only one correct answer to a multi-step mathematical expression. Without this hierarchy, a single equation could yield multiple results depending on which operation you choose to perform first. Think of it as the 'traffic rules' for arithmetic; just as the 5/20 rule once governed the flight paths of Indian carriers Indian Economy, Nitin Singhania .(ed 2nd 2021-22), Infrastructure, p.458, VBODMAS governs the flow of calculation to prevent logical collisions.
The acronym stands for:

• V (Vinculum or Bar): Represented by a horizontal line over numbers (e.g., ‾8-5‾). This has the absolute highest priority.
• B (Brackets): Solved in the order of Round (), Curly {}, and then Square [].
• O (Of / Orders): 'Of' is a specialized multiplication that takes precedence over division. 'Orders' refers to powers (exponents like x²) and roots (√). When evaluating algebraic expressions, such as those used to find market equilibrium Microeconomics (NCERT class XII 2025 ed.), Market Equilibrium, p.74, handling these powers correctly is vital.
• D & M (Division & Multiplication): These hold equal rank and should be performed from left to right as they appear in the expression.
• A & S (Addition & Subtraction): These also hold equal rank and are performed from left to right.

A common trap for intermediate learners is the 'Of' vs. 'Multiplication' distinction. For instance, in the expression 10 ÷ 2 of 5, the 'of' must be calculated first (2 × 5 = 10), resulting in 10 ÷ 10 = 1. If you mistakenly treated it as simple multiplication, you might incorrectly divide first. Similarly, when substituting negative values into a polynomial (like replacing x with -2), the 'Orders' rule dictates that you must square or cube the number including its sign before proceeding to the addition or subtraction phases. For example, (-2)³ becomes -8, and subtracting a negative ( -(-2) ) turns into addition (+2). Mastering these nuances prevents the 'silly mistakes' that often cost candidates precious marks in the aptitude paper.

Sources: Indian Economy, Nitin Singhania .(ed 2nd 2021-22), Infrastructure, p.458; Microeconomics (NCERT class XII 2025 ed.), Market Equilibrium, p.74

5. Remainder and Factor Theorems (intermediate)

In quantitative aptitude, the Remainder Theorem and the Factor Theorem are indispensable tools for solving polynomial problems without resorting to tedious long division. At its core, the Remainder Theorem states that if a polynomial f(x) is divided by a linear divisor of the form (x - a), the remainder is simply the value of the polynomial when a is substituted for x, denoted as f(a). This is rooted in the basic division algorithm: Dividend = (Divisor × Quotient) + Remainder. When the divisor is (x - a), the equation becomes f(x) = (x - a)q(x) + R. By substituting x = a, the term (x - a) becomes zero, leaving us with f(a) = R. Just as in governance where we calculate the remainder of a term for a dissolved Panchayat to determine election necessity Indian Polity, M. Laxmikanth(7th ed.), Panchayati Raj, p.389, in mathematics, we isolate the remainder to understand the "leftover" value of a function.

The Factor Theorem is a specialized application of this rule. It posits that (x - a) is a factor of the polynomial f(x) if and only if the remainder f(a) = 0. This is a critical logic gate for UPSC aspirants: if substituting a value results in zero, you have found a root of the equation. This process of substitution is similar to how we plug values of current and potential difference into Ohm’s Law to find resistance Science, class X (NCERT 2025 ed.), Electricity, p.193. By evaluating the behavior of a function at a specific point, we unlock its structural components.

Let us look at a practical evaluation. Suppose we have a cubic polynomial x³ - x² - x - 1 and we wish to find the remainder when it is divided by (x + 2). Instead of division, we substitute x = -2. Step-by-step: (-2)³ = -8 and (-2)² = 4. Plugging these in gives -8 - 4 - (-2) - 1. Simplifying further: -8 - 4 + 2 - 1 = -11. Because the result is not zero, we know -11 is the remainder and (x + 2) is not a factor of this polynomial.

Sources: Indian Polity, M. Laxmikanth(7th ed.), Panchayati Raj, p.389; Science, class X (NCERT 2025 ed.), Electricity, p.193

6. Evaluating Expressions by Substitution (intermediate)

Evaluating expressions by substitution is the process of replacing a variable (like x, p, or r) with a specific numerical value to find the final result. This concept is the backbone of quantitative analysis across all subjects. For instance, in economics, you might substitute a specific price into a demand function to find the equilibrium quantity Microeconomics (NCERT class XII 2025 ed.), Market Equilibrium, p.74, or substitute constants into a formula to derive the money multiplier Macroeconomics (NCERT class XII 2025 ed.), Money and Banking, p.51.

The most critical aspect of substitution, especially at an intermediate level, is the careful handling of signs and exponents. When substituting a negative number into a polynomial, it is a standard rule to place that number inside parentheses. This prevents common arithmetic errors involving the Order of Operations (BODMAS/PEMDAS). Remember these two fundamental rules for negative bases:

• A negative number raised to an even power results in a positive value (e.g., (-2)² = 4).
• A negative number raised to an odd power remains negative (e.g., (-2)³ = -8).

To evaluate a complex expression like x³ - x² - x - 1 for x = -2, follow a stepwise approach. First, replace every instance of x with (-2): (-2)³ - (-2)² - (-2) - 1. Next, calculate the powers: -8 - (4) - (-2) - 1. Then, simplify the signs (remembering that subtracting a negative is the same as adding a positive): -8 - 4 + 2 - 1. Finally, sum the terms from left to right: -12 + 2 - 1 = -10 - 1 = -11.

Sources: Microeconomics (NCERT class XII 2025 ed.), Market Equilibrium, p.74; Macroeconomics (NCERT class XII 2025 ed.), Money and Banking, p.51

7. Solving the Original PYQ (exam-level)

This question perfectly synthesizes the core principles of polynomial evaluation and integer arithmetic you just mastered. To solve it, you must apply the rules of exponents, specifically noting how negative bases behave: an odd power maintains the negative sign, while an even power results in a positive value. By treating the variable X as a placeholder for the value -2, you are essentially performing a systematic substitution that tests your precision with sign conventions, a fundamental skill emphasized in Mathcentre: Cubic Equations.

Let’s walk through the logic: first, substitute -2 into the expression X³ - X² - X - 1. Calculating the individual components gives us (-2)³ which is -8, and (-2)² which is 4. When you plug these back into the original string, the expression becomes -8 - (4) - (-2) - 1. The most critical step here is the double negative: subtracting -2 turns into adding 2. This leaves you with -8 - 4 + 2 - 1. Combining the negative terms gives you -13, and adding 2 results in -11, making (C) the correct choice. Slow down during the sign conversion phase, as this is where most errors occur.

UPSC often includes options that mirror common calculation traps to test your focus. For instance, if you incorrectly assumed that (-2)³ was positive 8, or if you failed to change the sign of the -X term to positive 2, you would likely land on distractors like (A) or (B). Option (D) -15 is a classic trap for students who correctly square and cube the numbers but ignore the sign of the variable during the final summation. Precision in operational signs is what separates a successful candidate from the rest in these high-stakes CSAT problems.