UK K12 GCSE/A-Level: Year 12 A-Level Lower Sixth Mathematics - Pure Binomial Expansion

Learning Objectives

Upon completing this topic, students will be able to:

• Expand expressions of the form (x + y)^n using the binomial theorem
• Apply the binomial theorem to simplify and manipulate expressions
• Identify and use the appropriate form of the binomial theorem (e.g. (x + y)^n, (x - y)^n)
• Understand the concept of the binomial coefficient and its application in the binomial theorem
• Use the binomial theorem to solve problems involving algebraic expressions

Core Concepts

The binomial theorem is a powerful tool for expanding expressions of the form (x + y)^n, where n is a positive integer. It states that:

(x + y)^n =? (n choose k) x^(n-k) y^k

where-denotes the sum over all values of k from 0 to n, and (n choose k) is the binomial coefficient.

The binomial coefficient (n choose k) can be calculated using the formula:

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

where ! denotes the factorial function.

The binomial theorem can be extended to negative values of n, in which case it becomes:

(x + y)^(-n) =? (-n choose k) x^(n-k) y^k

The binomial theorem can also be applied to expressions of the form (x - y)^n, in which case the expansion is:

(x - y)^n =? (n choose k) x^(n-k) (-y)^k

Worked Examples

Example 1

Expand (2x + 3y)^4 using the binomial theorem.

Using the binomial theorem, we have:

(2x + 3y)^4 =? (4 choose k) (2x)^(4-k) (3y)^k

Evaluating the binomial coefficients and simplifying, we get:

(2x + 3y)^4 = 16x^4 + 96x^3y + 216x^2y^2 + 216xy^3 + 81y^4

Example 2

Simplify the expression (x + 2y)^(-3) using the binomial theorem.

Using the binomial theorem for negative values of n, we have:

(x + 2y)^(-3) =? (-3 choose k) x^(3-k) (2y)^k

Evaluating the binomial coefficients and simplifying, we get:

(x + 2y)^(-3) = 1/x^3 - 6/x^2y + 12xy - 8y^2/x

Common Misconceptions

• Students may confuse the binomial coefficient with the factorial function.
• Students may forget to evaluate the binomial coefficients correctly.
• Students may apply the binomial theorem to expressions that are not in the correct form (e.g. applying it to (x - y)^n without changing the sign of y).

Exam Tips

• Make sure to evaluate the binomial coefficients correctly.
• Check that the expression is in the correct form before applying the binomial theorem.
• Use the binomial theorem to simplify and manipulate expressions, rather than just expanding them.

MCQs

MCQ 1 [F]

What is the value of (4 choose 2)?

A) 4 B) 6 C) 12 D) 24

Correct answer: B) 6 Why the distractors fail: A) is too small, C) and D) are too large.

MCQ 2 [H]

What is the value of (x + y)^3 using the binomial theorem?

A) x^3 + 3x^2y + 3xy^2 + y^3 B) x^3 + 3x^2y - 3xy^2 + y^3 C) x^3 - 3x^2y + 3xy^2 + y^3 D) x^3 - 3x^2y - 3xy^2 + y^3

Correct answer: A) x^3 + 3x^2y + 3xy^2 + y^3 Why the distractors fail: B) and C) have incorrect signs, D) has an incorrect coefficient.

MCQ 3 [F]

What is the value of (2x + 3y)^2 using the binomial theorem?

A) 4x^2 + 12xy + 9y^2 B) 4x^2 + 6xy + 9y^2 C) 4x^2 - 6xy + 9y^2 D) 4x^2 + 12xy - 9y^2

Correct answer: A) 4x^2 + 12xy + 9y^2 Why the distractors fail: B) and C) have incorrect coefficients, D) has an incorrect sign.

MCQ 4 [H]

What is the value of (x - y)^(-2) using the binomial theorem?

A) 1/(x^2 + 2xy + y^2) B) 1/(x^2 - 2xy + y^2) C) x^2 + 2xy + y^2 D) x^2 - 2xy + y^2

Correct answer: A) 1/(x^2 + 2xy + y^2) Why the distractors fail: B) and D) have incorrect signs.

MCQ 5 [F]

What is the value of (x + y)^2 using the binomial theorem?

A) x^2 + 2xy + y^2 B) x^2 - 2xy + y^2 C) x^2 + 3xy + y^2 D) x^2 - 3xy + y^2

Correct answer: A) x^2 + 2xy + y^2 Why the distractors fail: B) and C) have incorrect coefficients, D) has an incorrect sign.

Short-answer questions

1. Expand (2x + 3y)^4 using the binomial theorem.
2. Simplify the expression (x + 2y)^(-3) using the binomial theorem.
3. What is the value of (4 choose 2)?
4. What is the value of (x + y)^3 using the binomial theorem?
5. What is the value of (x - y)^(-2) using the binomial theorem?