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 ↓


General Term


The binomial theorem is a powerful tool in algebra that allows us to expand expressions raised to exponents. In particular, it deals with the following types of expressions:

(a + b) n

Expanding such expressions to large powers by hand would be troublesome, and the binomial theorem provides a systematic way to do this with a formula.

The binomial theorem states that:

(a + b) n = Σ (n choose k) a n-k b k, k = 0 to n

where "Σ" refers to the sum of terms of every integer k from 0 to n, and "(n choose k)" is a binomial coefficient.

Understanding binomial coefficients

The binomial coefficient, represented as (n choose k), gives the number of ways to choose k elements from a set of n elements, regardless of the order. It is given by the formula:

(n choose k) = n! / (k!(n-k)!)

Here, "!" represents the factorial, which is the product of all positive integers up to that number.

Let's look at an example:

If n = 5 and k = 2, then:

(5 choose 2) = 5! / (2!(5-2)!) = (5x4x3x2x1) / (2x1x3x2x1) = 10

This means that there are 10 ways to choose 2 elements from a group of 5 elements.

General term of binomial expansion

The purpose of finding the general term is to identify a specific term in the expansion of a binomial expression. If we are given a binomial expression raised to a high power, it is not always necessary to expand the entire expression just to find a term. The general term helps in such cases.

The general term of the binomial expansion (a + b) n is given as follows:

T k+1 = (n choose k) a n-k b k

Here, T k+1 denotes the (k + 1)th term. Note how the index k is used in the binomial coefficients and the powers of a and b.

To understand how the term common is derived and used, consider:

If (a + b) n is being expanded and you want to know the fourth term.

You can use the general term formula:

For the fourth term, k = 3 (since indices start from 0).
T 4 = (choose n 3) A n-3 B 3

By entering your specific values you will get your word.

Example 1: Finding specific words

Suppose we want to find the 5th term in the expansion of (x + y) 8.

  1. Identify n = 8 and k = 4 because the term we are looking for is the 5th one.
  2. Apply the general term formula:
    3. T 5 = (8 choose 4) x 8-4 y 4
  3. Calculate the binomial coefficient:
    4. (8 choose 4) = 8! / (4!4!) = 70
  4. Therefore, the 5th term is:
    5. T 5 = 70 x 4 y 4

Viewing words in detail

t1 = n T 2 = (choose n 1) A n-1 B 1 T 3 = (choose n 2) a n-2 b 2 , T r = (choose n r-1) A n-r+1 B r-1 t n+1 = b n

The diagram above shows an expansion where each term T k+1 is represented sequentially along a line. The size of the circles can visually represent the coefficients in the terms.

Practice problems

Let's apply what we've learned by solving some practice problems.

Problem 1

Find the 7th term of (2x - y) 10.

  1. Use the general term: T k+1 = (n choose k) a n-k b k.
  2. Identify n = 10, a = 2x, b = -y, and k = 6 (for the 7th term).
  3. Calculate the binomial term:
    4. (10 choose 6) = 210
  4. Plug the general term into the formula:
    5. T 7 = 210 * (2x) 4 *(-y) 6
  5. Simplify the words:
    6. T 7 = 210 * 16x 4 * y 6 * (-1)
T 7 = -3360x 4 y 6
  6. The 7th term in the expansion is -3360x 4 y 6.

Problem 2

What is the coefficient of x 5 in the expansion of (x + 3) 8?

  1. Use the general term T k+1 = (8 choose k) a 8-k b k.
  2. We are finding the coefficient of x 5 So put 8-k = 5 and make k = 3.
  3. Calculate the binomial coefficient:
    4. (8 choose 3) = 56
  4. Apply the formula:
    5. T 4 = 56 * x 5 * 3 3
T 4 = 56 * x 5 * 27
  5. Solve for the coefficients:
    6. Coefficient = 1512
  6. Hence, the coefficient of x 5 in (x + 3) 8 is 1512.

Key insights and conclusions

The concept of a general term in the binomial theorem provides an explicit and efficient way to pick out any specific term in the expansion of a binomial expression, without having to expand the entire expression. The power of the binomial theorem lies not only in its mathematical beauty, but also in its wide range of applications in algebra, calculus, and beyond.

By understanding how to find a particular term and the role of binomial coefficients, you gain an important tool in algebra that can simplify complex problems and provide insight into mathematical patterns. This helps to build a solid foundation for further study in math and science.


Grade 11 → 1.3.2


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Expansion of Binomials
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