2.1 Algebraic Expressions, Polynomials, Powers & Radicals

2.1 Algebraic Expressions, Polynomials, Powers & Radicals

Algebraic manipulation forms the core operational foundation of the Bocconi Admission Test. Approximately half of the 24 mathematics questions rely on fast algebraic reductions, factoring polynomials, evaluating rational powers, or simplifying nested radicals under a strict pace of 90 seconds per question. Mastery of core algebraic identities, polynomial division, and exponent rules enables candidates to solve complex items swiftly while avoiding deliberate test traps.

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Polynomial Arithmetic and the Division Algorithm

A polynomial in one real variable $x$ is an expression of the form $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$, with real coefficients $a_i$ ($a_n \neq 0$) and non-negative integer degree $\deg(P) = n$.

Degree Properties

• Multiplication: The degree of a product is the sum of the degrees: $\deg(P \cdot Q) = \deg(P) + \deg(Q)$.
• Addition and Subtraction: The degree of a sum is bounded by $\deg(P \pm Q) \le \max(\deg P, \deg Q)$. If equal-degree terms cancel, the degree drops strictly below $\max(\deg P, \deg Q)$.
• Division Algorithm: For polynomials $P(x)$ and $D(x) \neq 0$, there exist unique polynomials $Q(x)$ (quotient) and $R(x)$ (remainder) such that: $P(x) = D(x) \cdot Q(x) + R(x) \quad \text{where } \deg(R) < \deg(D) \text{ or } R(x) = 0$

Remainder and Factor Theorems

• Remainder Theorem: When $P(x)$ is divided by a linear monic divisor $(x - c)$, the remainder is the constant $R = P(c)$.
• Factor Theorem: The binomial $(x - c)$ is a factor of $P(x)$ if and only if $P(c) = 0$. Thus, $c$ is a real root of $P(x) = 0$.

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Factoring Techniques: Core Taxonomy

Factoring decomposes polynomial expressions into products of irreducible factors, which is essential for simplifying rational expressions and identifying critical values.

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Laws of Integer and Rational Exponents

For non-zero real bases $a, b$ and real exponents $m, n$:

1. Product of Powers: $a^m \cdot a^n = a^{m+n}$
2. Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$
3. Power of a Power: $(a^m)^n = a^{m \cdot n}$
4. Power of a Product and Quotient: $(ab)^n = a^n b^n$, and $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
5. Zero Exponent: $a^0 = 1$ for all $a \neq 0$ (note: $0^0$ is undefined)
6. Negative Exponents: $a^{-n} = \frac{1}{a^n}$, and $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$

Rational Exponents and Radicals

For integer $m$ and positive integer $n \ge 2$: $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

• Odd Index ($n$ odd): $\sqrt[n]{a}$ is uniquely defined for all real $a \in \mathbb{R}$ (e.g., $\sqrt[3]{-8} = -2$).
• Even Index ($n$ even): Defined in $\mathbb{R}$ only when $a \ge 0$, representing the non-negative principal root.
• The Principal Root Trap: $\sqrt{x^2} = |x|$. Writing $\sqrt{x^2} = x$ is invalid when $x < 0$. For instance, $\sqrt{(-4)^2} = \sqrt{16} = 4 = |-4|$.

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Rationalizing Denominators

Rationalization eliminates radicals from denominators, transforming expressions into standard forms:

1. Monomial Radicals: $\frac{A}{\sqrt{b}} = \frac{A\sqrt{b}}{b}$, and $\frac{A}{\sqrt[n]{b^k}} = \frac{A\sqrt[n]{b^{n-k}}}{b}$.
2. Binomial Conjugates: Multiply numerator and denominator by the conjugate $(\sqrt{a} \mp \sqrt{b})$: $\frac{A}{\sqrt{a} + \sqrt{b}} = \frac{A(\sqrt{a} - \sqrt{b})}{(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b})} = \frac{A(\sqrt{a} - \sqrt{b})}{a - b}$
3. Cubic Binomials: Multiply by the quadratic factor from the sum or difference of cubes: $\frac{1}{\sqrt[3]{a} + \sqrt[3]{b}} = \frac{\sqrt[3]{a^2} - \sqrt[3]{ab} + \sqrt[3]{b^2}}{a + b}$

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Bocconi Traps and Algebraic Shortcuts

• Freshman's Dream Trap: $\sqrt{a^2 + b^2} \neq a + b$ and $(a + b)^n \neq a^n + b^n$. Powers and roots do not distribute over sums.
• Cancellation Trap: In fractions, cancel common multiplying factors, never added terms: $\frac{x^2 + 4}{x} = x + \frac{4}{x} \neq x + 4$.
• Symmetric Power Invariants: When $x + \frac{1}{x} = k$, compute higher powers using expansions:
  • $x^2 + \frac{1}{x^2} = \left(x + \frac{1}{x}\right)^2 - 2 = k^2 - 2$
  • $x^3 + \frac{1}{x^3} = \left(x + \frac{1}{x}\right)^3 - 3\left(x + \frac{1}{x}\right) = k^3 - 3k$
  • $x^4 + \frac{1}{x^4} = (k^2 - 2)^2 - 2$

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Worked Step-by-Step Examples

Example 1: Rational Expression Reduction

Simplify $E(x) = \frac{x^4 - 16}{x^3 - 2x^2 + 4x - 8}$ for all $x \neq 2$:

1. Factor numerator by difference of squares: $x^4 - 16 = (x^2 - 4)(x^2 + 4) = (x - 2)(x + 2)(x^2 + 4)$.
2. Factor denominator by grouping: $x^2(x - 2) + 4(x - 2) = (x - 2)(x^2 + 4)$.
3. Cancel common non-zero factors $(x - 2)(x^2 + 4)$: $E(x) = \frac{(x - 2)(x + 2)(x^2 + 4)}{(x - 2)(x^2 + 4)} = x + 2$

Example 2: Radical Conjugate Simplification

Evaluate $S = \frac{6}{\sqrt{5} - \sqrt{2}} - \frac{3}{\sqrt{5} + \sqrt{2}} - 3\sqrt{2}$:

1. Rationalize first fraction: $\frac{6(\sqrt{5} + \sqrt{2})}{5 - 2} = \frac{6(\sqrt{5} + \sqrt{2})}{3} = 2\sqrt{5} + 2\sqrt{2}$.
2. Rationalize second fraction: $\frac{3(\sqrt{5} - \sqrt{2})}{5 - 2} = \frac{3(\sqrt{5} - \sqrt{2})}{3} = \sqrt{5} - \sqrt{2}$.
3. Combine terms: $(2\sqrt{5} + 2\sqrt{2}) - (\sqrt{5} - \sqrt{2}) - 3\sqrt{2} = 2\sqrt{5} + 2\sqrt{2} - \sqrt{5} + \sqrt{2} - 3\sqrt{2} = \sqrt{5}$.

Example 3: Rational Exponent Evaluation

Compute $Q = \left(\frac{27}{8}\right)^{-2/3} + 16^{3/4} \cdot 8^{-1/3}$:

1. First term: $\left(\frac{27}{8}\right)^{-2/3} = \left(\frac{8}{27}\right)^{2/3} = \left(\sqrt[3]{\frac{8}{27}}\right)^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9}$.
2. Second term: $16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8$, and $8^{-1/3} = \frac{1}{\sqrt[3]{8}} = \frac{1}{2}$. Their product is $8 \cdot \frac{1}{2} = 4$.
3. Sum components: $Q = \frac{4}{9} + 4 = \frac{40}{9}$.