Lesson Notes By Weeks and Term v3 - Senior Secondary 2
Binomial expansion
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Binomial expansion
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Subject: Further Mathematics
Class: Senior Secondary 2
Term: 1st Term
Week: 1
Theme: Pure Mathematics
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Construct the pascal triangle Write out the binomial expansion for (a+b)n for positive in teger, negative in tegers and fractional values Apply the binomial expansion to evaluating power of number e.g (1.06)
Teacher Activities: Introduction (10 minutes): Begin by recalling prior knowledge: Ask students to expand simple binomials like $(a+b)^2$ and $(a+b)^3$ manually.
Introduce the challenge: What if the power is much larger, e.g., $(a+b)^{10}$ or $(a+b)^{1/2}$? This sets the stage for the need for the binomial theorem. State the lesson objectives clearly.
Pascal's Triangle (15 minutes): Guide students through the construction of Pascal's Triangle row by row on the board, explaining the summation rule. Demonstrate how the coefficients in Pascal's Triangle relate to the expansion of $(a+b)^n$ for small 'n' using Worked Example
1. Have students practice constructing rows of the triangle up to $n=6$. Binomial Theorem for Positive Integers (25 minutes): Introduce the general formula $\binom{n}{r} = \frac{n!}{r!(n-r)!}$ and explain factorials. Show how to calculate binomial coefficients using this formula.
Present the general expansion formula: $(a+b)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r} b^r$. Work through Worked Example 2 and 3, emphasizing handling of negative terms and numerical coefficients. Introduce the formula for the $(r+1)^{th}$ term and demonstrate with Worked Example
4. Binomial Theorem for Negative and Fractional Powers (30 minutes): Explain that for non-positive integer powers, the expansion is an infinite series, and the standard formula changes to $(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \dots$ Crucially, highlight the condition for validity, $|x|<1$. Work through Worked Example 5 and 6, paying close attention to the sign changes and fractional arithmetic. Demonstrate how to handle expressions not in the $(1+x)^n$ form by factorisation (Worked Example 7). Application to Approximations (20 minutes): Explain how the first few terms of an expansion can provide accurate approximations when 'x' is small. Work through Worked Example 8 and 9, guiding students on how to identify 'x' and 'n' and perform calculations systematically. Emphasize the importance of decimal place accuracy.
Recap and Q&A (5 minutes): Summarize key concepts: Pascal's Triangle, general formula, generalized formula, validity conditions, and approximation technique. Address any student questions or areas of confusion.
Student Activities: Recall and Brainstorm: Students participate in expanding $(a+b)^2$ and $(a+b)^3$ and discussing the pattern.
Construction Practice: Individually or in pairs, students construct Pascal's Triangle up to Row 6 or
7. Cooperative Learning: Students work in small groups to expand binomials for positive integer powers using both Pascal's Triangle and the general formula.
Problem Solving: Students attempt to expand expressions with negative and fractional powers, ensuring they identify 'n' and 'x' correctly and state the conditions for validity.
Application Tasks: Students work on approximation problems, focusing on calculation accuracy and rounding to specified decimal places.
Presentation and Discussion: Selected students present their solutions to practice problems, explaining their steps. Others provide feedback and ask questions. The teacher should guide students through these questions, providing support and clarifications as needed.
Question 1: Construct Pascal's triangle up to Row
5. Use it to expand $(p-q)^4$.
Solution 1: Pascal's Triangle (Row 0 to Row 5): Row 0: 1 Row 1: 1 1 Row 2: 1 2 1 Row 3: 1 3 3 1 Row 4: 1 4 6 4 1 Row 5: 1 5 10 10 5 1 Expansion of $(p-q)^4$: Using Row 4 coefficients: $1, 4, 6, 4, 1$. Here $a=p$ and $b=-q$. $(p-q)^4 = 1 \cdot p^4(-q)^0 + 4 \cdot p^3(-q)^1 + 6 \cdot p^2(-q)^2 + 4 \cdot p^1(-q)^3 + 1 \cdot p^0(-q)^4$ $= p^4 - 4p^3q + 6p^2q^2 - 4pq^3 + q^4$ Question 2: Find the coefficient of $x^3$ in the expansion of $(3x+2)^5$.
Solution 2: The general term is $T_{r+1} = \binom{n}{r} a^{n-r} b^r$. Here $n=5$, $a=3x$, $b=2$. We want the term with $x^3$. In $a^{n-r} b^r$, the power of $x$ comes from $a^{n-r} = (3x)^{n-r} = (3x)^{5-r}$. So, $5-r=3 \implies r=2$. The term is $T_{2+1} = T_3 = \binom{5}{2} (3x)^{5-2} (2)^2$ $T_3 = \binom{5}{2} (3x)^3 (2)^2$ Calculate $\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = \frac{5 \times 4}{2 \times 1} = 10$. $T_3 = 10 \cdot (27x^3) \cdot (4)$ $T_3 = 10 \cdot 108x^3 = 1080x^3$. The coefficient of $x^3$ is $1080$.
Question 3: Expand $(1-3x)^{-1}$ up to the term in $x^3$, and state the range of values of $x$ for which the expansion is valid.
Solution 3: Using the generalized binomial theorem $(1+y)^n = 1 + ny + \frac{n(n-1)}{2!}y^2 + \frac{n(n-1)(n-2)}{3!}y^3 + \dots$ Here $n=-1$ and $y=-3x$. $(1-3x)^{-1} = 1 + (-1)(-3x) + \frac{(-1)(-1-1)}{2!}(-3x)^2 + \frac{(-1)(-1-1)(-1-2)}{3!}(-3x)^3 + \dots$ $= 1 + 3x + \frac{(-1)(-2)}{2}(9x^2) + \frac{(-1)(-2)(-3)}{6}(-27x^3) + \dots$ $= 1 + 3x + (1)(9x^2) + (-1)(-27x^3) + \dots$ $= 1 + 3x + 9x^2 + 27x^3 + \dots$ Condition for validity: $|-3x| < 1 \implies 3|x| < 1 \implies |x| < \frac{1}{3}$. So, $-\frac{1}{3} < x < \frac{1}{3}$.
Question 4: Find the first three terms in the expansion of $(8+12x)^{1/3}$ in ascending powers of $x$.
Solution 4: First, factor out 8 to get it in the form $(1+kx)^n$: $(8+12x)^{1/3} = [8(1+\frac{12x}{8})]^{1/3} = [8(1+\frac{3}{2}x)]^{1/3}$ $= 8^{1/3} (1+\frac{3}{2}x)^{1/3}$ $= 2 (1+\frac{3}{2}x)^{1/3}$ Now, expand $(1+\frac{3}{2}x)^{1/3}$ with $n=1/3$ and $y=\frac{3}{2}x$.
The first three terms are: $1 + ny + \frac{n(n-1)}{2!}y^2$ $= 1 + (\frac{1}{3})(\frac{3}{2}x) + \frac{(\frac{1}{3})(\frac{1}{3}-1)}{2!}(\frac{3}{2}x)^2 + \dots$ $= 1 + \frac{1}{2}x + \frac{(\frac{1}{3})(-\frac{2}{3})}{2}(\frac{9}{4}x^2) + \dots$ $= 1 + \frac{1}{2}x + \frac{-\frac{2}{9}}{2}(\frac{9}{4}x^2) + \dots$ $= 1 + \frac{1}{2}x + (-\frac{1}{9})(\frac{9}{4}x^2) + \dots$ $= 1 + \frac{1}{2}x - \frac{1}{4}x^2 + \dots$ Multiply the entire expansion by 2: $2(1 + \frac{1}{2}x - \frac{1}{4}x^2 + \dots) = 2 + x - \frac{1}{2}x^2 + \dots$ The first three terms are $2 + x - \frac{1}{2}x^2$.
Question 5: Expand $(1+x)^5$. Hence, evaluate $(1.02)^5$ correct to 3 decimal places.
Solution 5: Expand $(1+x)^5$: Using Pascal's Row 5 coefficients (1 5 10 10 5 1) or formula: $(1+x)^5 = \binom{5}{0}1^5x^0 + \binom{5}{1}1^4x^1 + \binom{5}{2}1^3x^2 + \binom{5}{3}1^2x^3 + \binom{5}{4}1^1x^4 + \binom{5}{5}1^0x^5$ $= 1 + 5x + 10x^2 + 10x^3 + 5x^4 + x^5$ Evaluate $(1.02)^5$: Here, $x=0.02$.
Substitute into the expansion: $(1.02)^5 = 1 + 5(0.02) + 10(0.02)^2 + 10(0.02)^3 + 5(0.02)^4 + (0.02)^5$ $= 1 + 0.10 + 10(0.0004) + 10(0.000008) + 5(0.00000016) + 0.0000000032$ $= 1 + 0.10 + 0.0040 + 0.000080 + 0.00000080 + 0.0000000032$ $= 1.1040808032$ Correct to 3 decimal places: $1.104$.
Differentiation: Tiered Activities: Provide different levels of complexity for practice problems. Basic expansions for some, complex coefficients/terms for others.
Peer Tutoring: Pair stronger students with those needing support to encourage collaborative learning.
Visual Aids: For visual learners, use colourful diagrams for Pascal's triangle and step-by-step flowcharts for the expansion process.
Remediation: Review Prerequisites: Revisit factorials and combinations ($\binom{n}{r}$) for students struggling with coefficient calculations. Conduct a quick mini-lesson on these basics.
Step-by-Step Breakdown: For initial expansions, guide struggling students through each step (identifying 'a', 'b', 'n'; applying powers; calculating coefficients; combining terms) with simpler examples (e.g., $(x+2)^3$).
Focused Practice on Pascal's Triangle: Provide extra practice worksheets solely on constructing Pascal's Triangle and using it for expansions up to $n=4$.
Error Analysis: Encourage students to identify common mistakes (e.g., sign errors with negative terms, incorrect application of powers to coefficients like $(2x)^3 \neq 2x^3$).
Extension/Enrichment: Binomial Distribution: Introduce the concept of binomial probability distribution in statistics, showing how the binomial coefficients are used to calculate probabilities of success in a series of independent trials (e.g., probability of getting a certain number of heads in coin tosses).
Series Convergence: For high-achieving students, explore the conditions for convergence of the generalized binomial series more deeply, including graphical representations of the approximation accuracy for different values of 'x'.
Alternative Applications: Task students to research other real-world applications of binomial expansion in fields like actuarial science, quantum mechanics, or signal processing. They could present their findings to the class. A binomial expression is an algebraic expression containing two terms connected by an addition or subtraction sign, e.g., $(a+b)$, $(x-2y)$. The Binomial Theorem provides a systematic way to expand expressions of the form $(a+b)^n$ for any positive integer $n$, and later extended for negative integers and fractional values.
Compound Interest Calculations (Finance): The formula for compound interest is $A = P(1+r)^t$, where A is the amount, P is the principal, r is the interest rate, and t is time. When $r$ is a small decimal, binomial expansion can be used to approximate the future value of an investment or a loan. Local
Example: A Nigerian saving N100,000 in a fixed deposit account offering 6% annual interest for 3 years. Using $(1+0.06)^3 \approx 1 + 3(0.06) + 3(0.06)^2 + (0.06)^3$, allows for a quick estimation of the total amount without a calculator, especially useful in rural banking or small business scenarios. Population Growth and Economic Forecasting: Population growth or economic indicators often follow exponential patterns that can be approximated by binomial expansions for small changes. For example, if Nigeria's population grows by a small percentage annually, say 2%, the binomial expansion $(1+0.02)^n$ can be used to predict population figures over a few years, particularly in initial stages or for short-term forecasts. This is useful for government planning agencies like the National Population Commission or National Bureau of Statistics. Error Analysis and Approximations in Science/Engineering: In physics or engineering, certain measurements might have small percentage errors. When these measurements are used in formulas involving powers, binomial expansion helps estimate the overall error without complex calculus. For example, in calculating the volume of a sphere with a radius measurement having a small error, $(r+\delta r)^3 \approx r^3 + 3r^2\delta r$, which can be simplified using binomial expansion. This could be applied in local manufacturing or construction projects to estimate tolerance levels or material requirements.