IB Mathematics AA SL Binomial theorem Study Notes
IB Mathematics AA SL Binomial theorem Study Notes Offer a clear explanation of Binomial theorem , including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Binomial theorem
Binomial Expansion
The Binomial Theorem provides a quick way of expanding expressions that are raised to a power, such as \((a + b)^n\). This theorem helps us see how terms are structured and how to calculate each coefficient using Pascal’s Triangle.
Understanding Pascal’s Triangle
Pascal’s Triangle is a triangular array that helps us identify the coefficients in a binomial expansion. For instance, the coefficients for each power of the expansion are found in the rows of Pascal’s Triangle, as shown below:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Each entry represents the number of ways to choose elements, or “combinations,” from a set. For example, \(C_4^2\) (4 choose 2) equals 6, as shown in the 4th row and 2nd column of Pascal’s Triangle.
Calculator Method for Combinations
- Type the total number (n).
- Select MATH, then PRB (probability), and choose nCr (combinations).
- Input the second number (r) and press ENTER.
In the example above, to find \(4 \text{ choose } 2\), enter 4 nCr 2 to get 6. This is read as “4 choose 2.”
Binomial Expansion Formula
The formula for the expansion of \((a + b)^n\) is:
\((a + b)^n = a^n + \binom{n}{1} a^{n-1}b + \binom{n}{2} a^{n-2}b^2 + \ldots + b^n\)
The sum of the powers of \(a\) and \(b\) in each term should equal the total power \(n\). Each coefficient can be found using Pascal’s Triangle or the combination formula.
Example of Binomial Expansion
Let’s expand \((2x – 3)^3\):
\((2x – 3)^3 = \binom{3}{0}(2x)^3(-3)^0 + \binom{3}{1}(2x)^2(-3)^1 + \binom{3}{2}(2x)^1(-3)^2 + \binom{3}{3}(2x)^0(-3)^3\)
Solution Steps
- Calculate each binomial coefficient using the formula or Pascal’s Triangle.
- Substitute the values and simplify to find each term.
Example Challenge Question
In the expansion of \((2x + 1)^n\), the coefficient of the \(x^3\) term is 80. Find the value of \(n\).
Solution:
Using the formula, we write the equation for the coefficient of \(x^3\):
\(\binom{n}{3}(2x)^3(1)^{n-3} = 80\).
This simplifies to \(\binom{n}{3} \cdot 8x^3 = 80x^3\), so \(\binom{n}{3} \cdot 8 = 80\).
Dividing, we get \(\binom{n}{3} = 10\).
From Pascal’s Triangle, we find that 10 appears in the 5th row, so \(n = 5\).
Find the coefficient of \(x^4\) in \((2x + 3)^6\)
Solution:
To find the coefficient of \(x^4\) in \((2x + 3)^6\), we use the binomial expansion formula:
\( ( 2x + 3 )^6 = \sum_{k=0}^{6} \binom{6}{k} (2x)^k 3^{6-k} \)
The term for \(x^4\) occurs when \(k=4\). Substituting \(k=4\) into the formula:
\(\binom{6}{4} (2x)^4 3^{6-4} = \binom{6}{4} (2x)^4 3^2 \)
Now, calculate:
\(\binom{6}{4} = \frac{6 \times 5}{2 \times 1} = 15 \)
\( (2x)^4 = 16x^4 \)
\( 3^2 = 9 \)
Thus, the term is:
\(15 \times 16 x^4 \times 9 = 2160 x^4\)
Therefore, the coefficient of \(x^4\) is 2160.
Properties of the Binomial Theorem
1. Symmetry in Binomial Coefficients
In a binomial expansion, the coefficients are symmetric. This means:
\(\binom{n}{r} = \binom{n}{n-r}\)
This property is evident in Pascal’s Triangle, where the coefficients on each row are symmetrical around the center.
2. Sum of Binomial Coefficients
The sum of all the coefficients in the expansion of \((a + b)^n\) is \(2^n\):
\((1 + 1)^n = 2^n\)
This result holds because the binomial theorem expansion is equivalent to setting \(a = 1\) and \(b = 1\) in the expression \((a + b)^n\).
3. Coefficient of Middle Term(s)
If \(n\) is even, there is one middle term in the expansion of \((a + b)^n\), which is given by:
\(\binom{n}{n/2}a^{n/2}b^{n/2}\)
If \(n\) is odd, there are two middle terms, corresponding to the terms with \(\binom{n}{(n-1)/2}\) and \(\binom{n}{(n+1)/2}\).
4. Exponent Sum Property
In each term of the expansion of \((a + b)^n\), the sum of the exponents of \(a\) and \(b\) is always equal to \(n\).
For example, in the expansion of \((a + b)^5\), each term has a total exponent sum of 5, such as \(a^5\), \(a^4b\), \(a^3b^2\), etc.
5. Special Cases in Binomial Expansion
- Case 1: When \(a = 1\), the expansion of \((1 + b)^n\) simplifies to a series of powers of \(b\) with coefficients given by the binomial coefficients.
- Case 2: When \(b = -1\), the expansion of \((a – 1)^n\) results in alternating signs for each term, depending on whether the power of \(b\) is odd or even.
6. Application to Approximations
The Binomial Theorem can be used to approximate expressions for values of \(a\) and \(b\) that are close to each other. For small values of \(b\) compared to \(a\), the initial terms in the expansion of \((a + b)^n\) provide an approximate value.
7. Derivation of Factorial-Based Coefficients
The binomial coefficient \(\binom{n}{r}\) can be expressed using factorials:
\(\binom{n}{r} = \frac{n!}{r!(n – r)!}\)
This formula is essential for calculating specific terms in a binomial expansion.
Additional Notes
- The Binomial Theorem plays a significant role in probability, combinatorics, and algebra.
- It helps simplify calculations involving powers of binomials and provides insights into series expansions.
IB Mathematics AA SL Binomial Theorem Exam Style Worked Out Questions
Question
Consider the binomial expansion \((1+x)^7=x^7+ax^6+bx^5+35x^4+..+1 where x\neq 0 and a,b \in \mathbb{Z}^+\)
(a) Show that b = 21 . [2]
The third term in the expansion is the mean of the second term and the fourth term in the expansion.
(b) Find the possible values of x .
▶️Answer/Explanation
Ans
Question
The values in the fourth row of Pascal’s triangle are shown in the following table.
Write down the values in the fifth row of Pascal’s triangle.
Hence or otherwise, find the term in \({x^3}\) in the expansion of \({(2x + 3)^5}\).
▶️Answer/Explanation
Markscheme
1, 5, 10, 10, 5, 1 A2 N2
[2 marks]
evidence of binomial expansion with binomial coefficient (M1)
eg\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} n \\ r \end{array}} \right){a^{n – r}}{b^r}\), selecting correct term, \({(2x)^5}{(3)^0} + 5{(2x)^4}{(3)^1} + 10{(2x)^3}{(3)^2} + \ldots \)
correct substitution into correct term (A1)(A1)(A1)
eg\(\,\,\,\,\,\)\(10{(2)^3}{(3)^2},{\text{ }}\left( {\begin{array}{*{20}{c}} 5 \\ 3 \end{array}} \right){(2x)^3}{(3)^2}\)
Note: Award A1 for each factor.
\(720{x^3}\) A1 N2
Notes: Do not award any marks if there is clear evidence of adding instead of multiplying.
Do not award final A1 for a final answer of 720, even if \(720{x^3}\) is seen previously.
[5 marks]