Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Algebra → Binomial Theorem ↓


Expansion of Binomials


In algebra, expanding binomials is a method used to expand expressions that are raised to a power or exponent. A binomial is an algebraic expression that has exactly two terms. These terms can be constants, variables, or a mixture of both. The binomial theorem provides a formula that describes the algebraic expansion of powers of a binomial.

The general form of a binomial raised to a power is:

(a + b) n

where a and b are individual terms of the binomial, and n is a non-negative integer representing the power or powers to which the binomial is raised.

Binomial theorem

The binomial theorem states that for any positive integer n, the expression (a + b) n can be expanded as follows:

(a + b) n = C(n, 0) * a n * b 0 + C(n, 1) * a n-1 * b 1 + ... + C(n, n-1) * a 1 * b n-1 + C(n, n) * a 0 * b n

Or, more succinctly:

(a + b) n = Σ (C(n, k) * a nk * b k ), where k ranges from 0 to n

C(n, k) refers to the binomial coefficient, which can be calculated by the following formula:

C(n, k) = n! / (k!(nk)!)

where n! denotes the factorial of n.

Understanding binomial coefficients

An important aspect of expanding binomials is understanding the binomial coefficients, which determine the weights of the terms in the expansion. Binomial coefficients can be found in Pascal's triangle, an arrangement where each entry is the sum of the two entries immediately above it.

111121133114641

Each row of Pascal's triangle corresponds to the coefficients of the expanded form for the increasing powers of the binomial. For example, the third row represents the coefficients of 1, 2, 1 (a + b) 2, and so on.

Examples of binomial expansion

To understand the process of binomial expansion more clearly, let us look at some examples.

Example 1: Expand (x + y) 3

To expand this binomial, we use the formula:

(x + y) 3 = C(3, 0) * x 3 * y 0 + C(3, 1) * x 2 * y 1 + C(3, 2) * x 1 * y 2 + C(3, 3) * x 0 * y 3

From Pascal's triangle or using the binomial coefficient formula, we get:

  • C(3, 0) = 1
  • C(3, 1) = 3
  • C(3, 2) = 3
  • C(3, 3) = 1

Re-substituting these coefficients into the expression, we get:

(x + y) 3 = 1 * x 3 + 3 * x 2 * y + 3 * x * y 2 + 1 * y 3

So, the expanded form is:

x 3 + 3x 2 y + 3xy 2 + y 3
x 33x 2 years3xy 2Y 3

Example 2: Expand (2a - 3b) 4

In this example, note the difference in signs between the two terms.

The formula is as follows:

(2a - 3b) 4 = C(4, 0) * (2a) 4 * (-3b) 0 + C(4, 1) * (2a) 3 * (-3b) 1 + ... + C(4, 4) * (2a) 0 * (-3b) 4

Using the coefficients of Pascal's triangle:

  • C(4, 0) = 1
  • C(4, 1) = 4
  • C(4, 2) = 6
  • C(4, 3) = 4
  • C(4, 4) = 1

Calculating each term independently gives:

= 1 * (2a) 4 + 4 * (2a) 3 * (-3b) + 6 * (2a) 2 * (-3b) 2 + 4 * (2a) * (-3b) 3 + 1 * (-3b) 4

Evaluating these gives:

= 1 * 16a 4 - 96a 3 b + 216a 2 b 2 - 216ab 3 + 81b 4

The expansion of which is as follows:

16a 4 - 96a 3 b + 216a 2 b 2 - 216ab 3 + 81b 4
16A 4-96A3B216A 2 B 2-216AB381B 4

Important considerations

It is important to keep track of signs and coefficients when expanding binomials, especially when negative numbers are involved. The sum of the exponents of each term should always be n, maintaining alternating order between a and b.

More practice

Try expanding some binomials yourself for a better understanding:

  • (3x + 2y) 2
  • (5 m - u) 3
  • (U+V) 4

Use the binomial theorem as your guide, calculate the coefficients with simple arithmetic, and practice constructing expanded polynomials accurately. Refer to Pascal's triangle for questions on binomial coefficients if necessary.

Conclusion

Understanding the expansion of binomials is a core element in algebra that shows the relationships between terms raised to exponents, while demonstrating the pattern of coefficients described by the binomial coefficients. With practice, the structure of expanded binomials becomes simpler to recognize, and taking advantage of this leads to a broader understanding in polynomial math.


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Expansion of Binomials

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