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Algebra 2 Identities

A list of essential Algebra 2 identities and formulas that are widely used:

1. Polynomial Identities

Difference of Squares

\[ a^2 - b^2 = (a - b)(a + b) \]

Perfect Square Trinomials

  • Positive:
\[ a^2 + 2ab + b^2 = (a + b)^2 \]
  • Negative:
\[ a^2 - 2ab + b^2 = (a - b)^2 \]

Sum and Difference of Cubes

  • Sum of Cubes:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
  • Difference of Cubes:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

2. Quadratic Identities

Quadratic Formula

For any quadratic equation of the form \( ax^2 + bx + c = 0 \):

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Discriminant

The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c = 0 \) is:

\[ \Delta = b^2 - 4ac \]
  • If \( \Delta > 0 \), there are two distinct real roots.
  • If \( \Delta = 0 \), there is one real root.
  • If \( \Delta < 0 \), there are two complex roots.

3. Logarithmic and Exponential Identities

Properties of Logarithms

  • Product Rule:
\[ \log_b(xy) = \log_b(x) + \log_b(y) \]
  • Quotient Rule:
\[ \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \]
\[ \log_b(x^n) = n \log_b(x) \]
  • Change of Base Formula:
\[ \log_b(x) = \frac{\log_a(x)}{\log_a(b)} \]

Exponential Properties

  • Product of Powers:
\[ a^m \times a^n = a^{m+n} \]
  • Quotient of Powers:
\[ \frac{a^m}{a^n} = a^{m-n} \]
  • Power of a Power:
\[ (a^m)^n = a^{mn} \]

Exponential and Logarithmic Inverses

  • Exponential and Logarithmic Relation:
\[ \log_b(b^x) = x \quad \text{and} \quad b^{\log_b(x)} = x \]

4. Rational Expressions and Identities

Reciprocal Identity

\[ \frac{1}{\frac{a}{b}} = \frac{b}{a} \]

Complex Fraction Identity

\[ \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc} \]

5. Absolute Value Identities

Absolute Value Properties

  • Multiplication:
\[ |ab| = |a| \cdot |b| \]

  • Division: [ \left|\frac{a}{b}\right| = \frac{|a|}{|b|}, \quad b \neq 0 ]

  • Absolute Value of a Product: [ |a^2| = a^2 ]

6. Complex Number Identities

Complex Conjugates

For a complex number \( z = a + bi \):

  • Conjugate: \( \overline{z} = a - bi \)
  • Product of a Complex Number and Its Conjugate:
\[ z \cdot \overline{z} = a^2 + b^2 \]

7. Binomial Theorem

The binomial expansion of \( (a + b)^n \) is given by:

\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]

where \( \binom{n}{k} \) is the binomial coefficient \( \frac{n!}{k!(n-k)!} \).

8. Sum and Product of Roots (Quadratic Equations)

For a quadratic equation \( ax^2 + bx + c = 0 \), the sum and product of the roots are:

  • Sum of Roots: [ r_1 + r_2 = -\frac{b}{a} ]

  • Product of Roots:

\[ r_1 \times r_2 = \frac{c}{a} \]

9. Rational Root Theorem

For a polynomial \( P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_0 \), any rational solution \( \frac{p}{q} \) is such that:

  • \( p \) is a factor of the constant term \( a_0 \).
  • \( q \) is a factor of the leading coefficient \( a_n \).

10. Partial Fraction Decomposition

To decompose a rational expression into simpler fractions:

\[ \frac{P(x)}{Q(x)} = \frac{A}{(x - r_1)} + \frac{B}{(x - r_2)} \]

where \( r_1 \) and \( r_2 \) are the roots of the denominator \( Q(x) \).