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


The binomial theorem is an important formula in algebra that describes the algebraic expansion of powers of binomials, which are expressions of the form (a + b) In simple terms, the binomial theorem tells us that for any positive integer n (a + b) n How to expand expressions of the form (a + b) n. This theorem is very useful in algebra, combinatorics and calculus.

Understanding the basics

A binomial is an algebraic expression that has exactly two terms. For example, (x + y), (3 + 4) or (a - b) are all binomials. Now, if we want to raise a binomial to a power, then we use the binomial theorem.

Formula

The binomial theorem states that:

    (a + b) n = ∑ (n choose k) * a n-k * b k 
       = c(n, 0)a n + c(n, 1)a n-1 b + c(n, 2)a n-2 b 2 + ... + c(n, n)b n

In the above formula:

  • n is a non-negative integer.
  • (a + b) n is called the binomial expansion.
  • C(n, k), also written (n choose k), is a binomial coefficient.
  • The symbol ∑ is the sum sign, which means we add multiple terms.

Understanding binomial coefficients

Binomial coefficients, denoted as C(n, k) or (n choose k), are the coefficients found in the expansion. They are given by the formula:

    C(n, k) = n! / (k! * (n - k)!)

Here, n! is the factorial of n, which is the product of all positive integers up to n. For example, 4! = 4 * 3 * 2 * 1 = 24.

Examples of binomial expansion

Example 1: Expansion of (x + y) 2

Let's use the binomial theorem to expand (x + y) 2.

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

    = 1 * x 2 + 2 * x * y + 1 * y 2

    = x 2 + 2xy + y 2

Example 2: (a + b) 3 expansion

Now expand (a + b) 3.

    (a + b) 3 = c(3,0) * a 3 * b 0 + c(3,1) * a 2 * b 1 
    + c(3,2) * a 1 * b 2 + c(3,3) * a 0 * b 3

    = 1 * a 3 + 3 * a 2 b + 3 * a b 2 + 1 * b 3

    = a 3 + 3a 2 b + 3ab 2 + b 3

Visual example

To further understand how the binomial expansion works, let's look at a visual example using simple graphics:


    
    x 2
    
    
    2xy
    
    
    y 2

The above visual example shows the expansion of (x + y) 2 = x 2 + 2xy + y 2.

Relation to Pascal's triangle

The coefficients of the terms in the binomial expansion correspond to numbers in Pascal's triangle. Each row of Pascal's triangle represents the coefficients of the expansion of a binomial expression. For example, the fourth row represents the coefficients of the binomial expansion of 1, 3, 3, 1 (a + b) 3. corresponds to the coefficients of the expansion.

Understanding Pascal's triangle

The triangle starts with a 1 at the top, and each row is constructed by adding the number directly above and to the left with the number directly above and to the right, with blank entries considered as 0. Let's look at the first few rows. See:

         1
        1 1
       1 2 1
      1 3 3 1
     1 4 6 4 1

Each row gives the coefficients for the expansion of n (a + b) n.

Example 3: Using Pascal's triangle for (x + 1) 4

Using the fifth row of Pascal's triangle 1, 4, 6, 4, 1, we can expand (x + 1) 4 to:

    (x + 1) 4 = 1 * x 4 + 4 * x 3 + 6 * x 2 + 4 * x 1 + 1 * x 0

    = x 4 + 4x 3 + 6x 2 + 4x + 1

Practical applications

The binomial theorem has many applications in mathematics, especially in algebra and calculus:

  • Calculus: The binomial theorem is used to find approximations and solve differential equations.
  • Probability theory: It is used to calculate the probability in the binomial distribution.
  • Combinatorics: Count the number of ways to select a subset of items from a larger set.
  • Algebraic identities: Important in simplifying expressions and solving algebraic problems.

Conclusion

The binomial theorem is a powerful tool in mathematics for expanding binomials to any power. Its applications extend beyond simple algebraic expansion to probability, calculus, and combinatorics, making it an important concept in mathematics. Theorem By using it and understanding its relation to Pascal's triangle, students can discover new ways to work with algebraic expressions and enhance their problem-solving skills.

Try it yourself!

Expand (m - 5) 3 using the binomial theorem.

Solution:

        (m - 5) 3 = C(3,0) * m 3 * (-5) 0 + C(3,1) * m 2 * (-5) 1 
        + c(3,2) * m 1 * (-5) 2 + c(3,3) * m 0 * (-5) 3

        = 1 * m 3 - 3 * m 2 * 5 + 3 * m * 25 - 1 * 125

        = m 3 – 15m 2 + 75m – 125

Grade 11 → 1.3


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Binomial Theorem

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