Edexcel A level Maths: (Part 2) 4.3 Using Partial Fractions In Binomial Expansion

Summary

This tutorial covers the method of partial fractions, beginning with expressing 8x+4 over (1-x)(x^2+x) as partial fractions. It guides through finding constants A and B, then expands the fraction in ascending powers of x. The video also discusses the validity of binomial expansions, determining the range of x values for which the expansions are valid. Lastly, it addresses improper fractions, performing polynomial long division, and expanding fractions up to the term in x squared using binomial expansion.

Takeaways

• 📚 The video tutorial focuses on part 2 of a 4-part series on using partial fractions.
• 🔍 The first example involves expressing \( \frac{8x + 4}{(1 - x)(x^2 + x)} \) as partial fractions.
• 📐 The process starts by setting up the equation with a common denominator and comparing numerators to form equations for constants A and B.
• 🔢 To find A, the tutorial uses substitution by setting \( x = 1 \), resulting in \( a = 4 \).
• 🔄 Similarly, to find B, \( x = -2 \) is substituted, yielding \( b = -4 \).
• 📝 For Part B, the fraction from Part A is expanded in ascending powers of X up to the term in X.
• 📈 The binomial expansion formula is applied to expand the terms, with careful attention to replacing variables as needed.
• 📉 Part C discusses the set of values for X for which the expansion is valid, considering the modulus of X.
• 🔑 The tutorial emphasizes the importance of checking if a fraction is improper before proceeding with partial fraction decomposition.
• 📐 For the second question, long division is used to handle an improper fraction, resulting in a quotient and a remainder.
• 🔎 Constants B and C are determined using substitution, with B found to be -2 and C as 3.

Q & A

What is the goal of the video tutorial?

The goal of the video tutorial is to explore partial fractions, focusing on expressing a given rational expression and using binomial expansion for the second part.

What is the initial expression presented in Part A?

The initial expression in Part A is (8x + 4) / ((1 - x)(2 + x)), and the goal is to express this as partial fractions.

How is the partial fraction decomposition set up in Part A?

The expression is set up as A / (1 - x) + B / (2 + x), where A and B are constants to be determined.

How is the value of A determined?

The value of A is determined by substituting x = 1, simplifying the equation to 12 = 3A, which gives A = 4.

How is the value of B determined?

The value of B is determined by substituting x = -2, which simplifies to -12 = 3B, giving B = -4.

What does the final partial fraction expression look like after finding A and B?

The final partial fraction expression is 4 / (1 - x) - 4 / (2 + x).

What is the next task in Part B of the question?

In Part B, the task is to expand the partial fractions in ascending powers of x using the binomial expansion.

Why is factorization required before applying the binomial expansion formula?

Factorization is required because the first term in the second fraction (2 + x) is not 1, so it must be rewritten as 2(1 + (1/2)x) to apply the binomial expansion.

What is the final expanded expression in Part B?

The final expanded expression after using the binomial expansion is 2 + 5x + (7/2)x².

What is the valid range of x for the expansion?

The valid range of x for the expansion is -1 < x < 1, determined by analyzing the conditions for the two binomial expansions used.