Key takeaway: The binomial theorem expands (a + b)ⁿ into a sum of terms C(n, k)·aⁿ⁻ᵏ·bᵏ, where the coefficients C(n, k) come from Pascal’s triangle.
Grade 12 – PrecalculusA-APR.C.5Expressions and Equations
Definition of Binomial Theorem
The binomial theorem is a pattern for multiplying out a two-term expression raised to a whole-number power, without multiplying it by itself over and over. The exponent of the first term counts down from \(n\) to 0 while the exponent of the second counts up from 0 to \(n\), so every term has total degree \(n\).
The numbers in front of each term, the binomial coefficients, count the ways to choose which factors contribute the second term. They can be read from Pascal’s triangle or computed with factorials. The expansion always has \(n+1\) terms, and the coefficients are symmetric from left to right.
Formula and notation
For a whole number \(n\):
\[(a+b)^n=\sum_{k=0}^{n}\binom\,{n}{k}a^{\,n-k}b^{\,k},\qquad \binom\,{n}{k}=\frac{n!}{k!\,(n-k)!}.\]
The term numbered \(k+1\) is \(\binom\,{n}{k}a^{n-k}b^{k}\). If a term is subtracted, treat it as a negative number inside \(b\).
Key facts
- The sum of all binomial coefficients in (a + b)ⁿ equals 2ⁿ.
- Setting a = b = 1 in the binomial theorem shows why the coefficients add to 2ⁿ.
- The coefficient pattern is symmetric: C(n, k) equals C(n, n − k).
- Subtraction in the binomial makes the signs alternate across the expansion.
Binomial Theorem example
Expand \((2x-3)^4\).
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Take \(a=2x\), \(b=-3\), \(n=4\), with coefficients 1, 4, 6, 4, 1.
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Terms: \(1(2x)^4=16x^4\); \(4(2x)^3(-3)=-96x^3\); \(6(2x)^2(-3)^2=216x^2\); \(4(2x)(-3)^3=-216x\); \((-3)^4=81\).
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Check at \(x=1\): \(16-96+216-216+81=1\), which equals \((2-3)^4=1\).
\((2x-3)^4=\)\(16x^4-96x^3+216x^2-216x+81\).
Common mistake
Watch out
Talia expands \((2x-3)^4\) with all plus signs: \(16x^4+96x^3+216x^2+216x+81\).
Why it is wrong
The second term of the binomial is negative, so it carries its sign into every power. Odd powers of \(-3\) are negative, which makes the terms with \(x^3\) and \(x\) negative. The all-plus result breaks the binomial theorem check at \(x=1\): it gives 625, not 1.
Related terms and lessons
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