Welcome to Binomial Expansions!
Have you ever looked at an expression like \((x + 2)^5\) and thought, "I really don't want to multiply those brackets out one by one"? You're not alone! Multiplying brackets over and over is time-consuming and it’s very easy to make a small mistake that ruins the whole answer.
In this chapter, we are going to learn a mathematical "shortcut" called the Binomial Expansion. This method allows us to expand brackets raised to any positive whole number power quickly and accurately. It’s a vital tool for your AS Level Maths journey and shows up everywhere from probability to calculus.
1. The Building Blocks: Factorials
Before we jump into the expansion, we need to understand a special piece of notation: the factorial.
The symbol for a factorial is an exclamation mark !. In maths, it doesn’t mean we are shouting the number; it means we multiply that number by every whole number below it, all the way down to 1.
Example:
\(4! = 4 \times 3 \times 2 \times 1 = 24\)
\(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)
Did you know?
There is one weird rule you just have to memorize: \(0! = 1\). It might seem strange, but it makes the formulas we are about to use work perfectly!
Quick Review: Factorials
• \(n!\) means \(n \times (n-1) \times (n-2) ... \times 1\)
• Most scientific calculators have an \(x!\) button to do this for you!
• Only use positive integers (whole numbers) for factorials in this section.
2. Choosing Objects: \(_nC_r\) and Combinations
The next tool in our kit is the combination notation, written as \(_nC_r\) or \(\binom{n}{r}\). This represents the number of ways to choose \(r\) items from a group of \(n\) items.
In binomial expansions, these values provide the "coefficients" (the numbers in front of the variables).
The formula for \(_nC_r\) is:
\(_nC_r = \frac{n!}{r!(n - r)!}\)
Don't worry if this seems tricky! You don't usually have to calculate this by hand. Look for the \(nCr\) button on your calculator (often found above the divide symbol or in a "Probability" menu).
Example:
How many ways can you choose 2 colors from a box of 5?
Using the formula: \(_5C_2 = \frac{5!}{2!(5-2)!} = \frac{120}{2 \times 6} = 10\).
Your calculator will give you 10 instantly if you type \(5\), then the \(nCr\) button, then \(2\).
Key Takeaway
The terms \(_nC_r\) (also written as \(\binom{n}{r}\)) tell us the numbers that appear in our expansion. For an expansion of power \(n\), we use \(_nC_0, _nC_1, _nC_2...\) all the way to \(_nC_n\).
3. Pascal’s Triangle: A Visual Shortcut
If you don't want to use the \(_nC_r\) formula, you can use Pascal’s Triangle. It’s a triangular array of numbers where each number is the sum of the two numbers directly above it.
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
The Trick: The numbers in Row \(n\) of Pascal's Triangle are exactly the same as the values of \(_nC_r\) for that power! For example, the numbers in Row 4 (\(1, 4, 6, 4, 1\)) are the values of \(_4C_0, _4C_1, _4C_2, _4C_3, _4C_4\).
4. The Binomial Expansion Formula
Now we put it all together. For the AS Level MEI syllabus, you need to be able to expand \((a + bx)^n\), where \(n\) is a positive integer.
The expansion follows a very strict pattern:
1. The numbers come from \(_nC_r\) (or Pascal's Triangle).
2. The powers of \(a\) start at \(n\) and decrease to 0.
3. The powers of \((bx)\) start at 0 and increase to \(n\).
The General Formula:
\((a + bx)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}(bx)^1 + \binom{n}{2}a^{n-2}(bx)^2 + ... + \binom{n}{n}(bx)^n\)
Step-by-Step Example: Expand \((2 + 3x)^3\)
Step 1: Find the coefficients for \(n=3\) from Pascal's Triangle or \(_nC_r\).
The numbers are: \(1, 3, 3, 1\).
Step 2: Set up the powers of \(a\) (which is 2).
They go down: \(2^3, 2^2, 2^1, 2^0\).
Step 3: Set up the powers of \(bx\) (which is \(3x\)).
They go up: \((3x)^0, (3x)^1, (3x)^2, (3x)^3\).
Step 4: Combine them.
Term 1: \(1 \times 2^3 \times (3x)^0 = 1 \times 8 \times 1 = 8\)
Term 2: \(3 \times 2^2 \times (3x)^1 = 3 \times 4 \times 3x = 36x\)
Term 3: \(3 \times 2^1 \times (3x)^2 = 3 \times 2 \times 9x^2 = 54x^2\)
Term 4: \(1 \times 2^0 \times (3x)^3 = 1 \times 1 \times 27x^3 = 27x^3\)
Final Answer:
\(8 + 36x + 54x^2 + 27x^3\)
5. Common Mistakes to Avoid
1. Forgetting brackets around \(bx\):
In the example above, many students write \(3x^2\) instead of \((3x)^2\). Remember that \((3x)^2 = 9x^2\). This is the most common reason for losing marks!
2. Negative signs:
If the bracket is \((a - bx)^n\), treat the second term as \((-bx)\). When you raise a negative number to an even power, it becomes positive. When you raise it to an odd power, it stays negative.
3. Power 0 and Power 1:
Always remember that anything to the power of 0 is 1 (e.g., \(2^0 = 1\)) and anything to the power of 1 is just itself.
Summary Table for \((a+b)^n\)
• Number of terms: There are always \(n+1\) terms.
• Sum of powers: In every single term, the powers of \(a\) and \(b\) must add up to \(n\).
• Symmetry: The coefficients are symmetrical (e.g., \(1, 4, 6, 4, 1\)).
6. Finding a Specific Term
Sometimes the exam doesn't want the whole expansion; it just wants one specific term, like the "coefficient of \(x^2\)".
To find the term containing \(x^r\) in the expansion of \((a + bx)^n\), use this piece of the formula:
Term = \(\binom{n}{r} \times a^{n-r} \times (bx)^r\)
Example: Find the coefficient of \(x^2\) in \((5 + 2x)^6\).
Here, \(n=6\), \(a=5\), \(bx=2x\), and we want \(r=2\).
Term = \(\binom{6}{2} \times 5^{6-2} \times (2x)^2\)
Term = \(15 \times 5^4 \times 4x^2\)
Term = \(15 \times 625 \times 4x^2 = 37,500x^2\).
The coefficient is 37,500.
Quick Review: Key Points
• \(\binom{n}{r}\) tells you the number of ways to choose \(r\) from \(n\).
• Powers of the first term go down, powers of the second term go up.
• Use brackets carefully when the second term involves a number and \(x\).
• The coefficient is just the number, don't include the \(x\) in your final answer if asked for the coefficient!