Binomial Expansion for Negative & Fractional Powers & Validity
In Year 1 Pure Maths, the binomial expansion of (a + b)^n only works when n is a positive integer, producing a finite polynomial of n + 1 terms using nCr coefficients. In Year 2 Pure Maths, Isaac Newton's general binomial theorem extends the expansion of (1 + x)^n to any rational power n, including negative integers (such as (1 + x)^(-2) = 1 / (1 + x)²) and fractions (such as (1 + x)^(1/2) = √(1 + x)). Because this infinite series only converges when |x| < 1, learning how to factor out a constant from (a + bx)^n and combining partial fractions with binomial expansions are top-tier exam skills.
- The general binomial expansion formula
- For any rational number n (negative or fractional) and |x| < 1, the infinite series expansion is: (1 + x)^n = 1 + nx + [n(n - 1) / 2!]x² + [n(n - 1)(n - 2) / 3!]x³ + . + [n(n - 1).(n - r + 1) / r!]x^r + . Notice that you cannot use the nCr button on your calculator when n is negative or fractional; you must substitute n directly into n(n - 1) / 2 and n(n - 1)(n - 2) / 6.
- Factoring (a + bx)^n into a^n (1 + (b/a)x)^n
- The general binomial formula only applies when the first term inside the bracket is 1. To expand (a + bx)^n where a ≠ 1, you must first factor a out of the bracket, remembering to raise a to the power n: (a + bx)^n = [a(1 + (b/a)x)]^n = a^n (1 + (b/a)x)^n. Expand (1 + (b/a)x)^n using X = (b/a)x (putting brackets around (b/a)x in every power term), and finally multiply every term inside by the outside factor a^n.
- Range of Validity (Convergence Interval)
- When n is a positive integer, the expansion of (a + bx)^n terminates and is valid for all real values of x (x ∈ ℝ). When n is negative or fractional, the series is infinite and converges if and only if the whole second term inside (1 + X)^n has modulus strictly less than 1: |X| < 1. Thus for (1 + kx)^n, it is valid for |kx| < 1 → |x| < 1 / |k|. For (a + bx)^n = a^n(1 + (b/a)x)^n, it is valid for |(b/a)x| < 1 → |x| < |a / b|.
Quadratic Parabola Plotter, Roots, Turning Point & Quadratic Formula
Adjust coefficients a, b, and c in y = ax² + bx + c to see how the parabola, roots (x-intercepts), and turning point move on the graph.
y = x² - 2x - 3
(-2)² − 4(1)(-3) = 16 → 2 real roots (crosses x-axis twice)x = 1.00b in brackets when squaring: (-2)² = 4 (always positive!). Typing -2² into a calculator without brackets gives -4 and loses the accuracy mark.Key Concepts & Exam Definitions (6)
- The general binomial expansion formula
- For any rational number n (negative or fractional) and |x| < 1, the infinite series expansion is: (1 + x)^n = 1 + nx + [n(n - 1) / 2!]x² + [n(n - 1)(n - 2) / 3!]x³ + . + [n(n - 1).(n - r + 1) / r!]x^r + . Notice that you cannot use the nCr button on your calculator when n is negative or fractional; you must substitute n directly into n(n - 1) / 2 and n(n - 1)(n - 2) / 6.
- Factoring (a + bx)^n into a^n (1 + (b/a)x)^n
- The general binomial formula only applies when the first term inside the bracket is 1. To expand (a + bx)^n where a ≠ 1, you must first factor a out of the bracket, remembering to raise a to the power n: (a + bx)^n = [a(1 + (b/a)x)]^n = a^n (1 + (b/a)x)^n. Expand (1 + (b/a)x)^n using X = (b/a)x (putting brackets around (b/a)x in every power term), and finally multiply every term inside by the outside factor a^n.
- Range of Validity (Convergence Interval)
- When n is a positive integer, the expansion of (a + bx)^n terminates and is valid for all real values of x (x ∈ ℝ). When n is negative or fractional, the series is infinite and converges if and only if the whole second term inside (1 + X)^n has modulus strictly less than 1: |X| < 1. Thus for (1 + kx)^n, it is valid for |kx| < 1 → |x| < 1 / |k|. For (a + bx)^n = a^n(1 + (b/a)x)^n, it is valid for |(b/a)x| < 1 → |x| < |a / b|.
- Combining Multiple Expansions & Intersecting Validity
- When an expression is the sum or product of two separate binomial factors, for example, (1 + 2x)^(-1) + (3 - x)^(-2) - each part has its own validity condition: |2x| < 1 → |x| < 1/2, and |-x/3| < 1 → |x| < 3. For the combined expansion to be valid, both individual expansions must converge simultaneously, so you must choose the stricter (smaller) interval: |x| < 1/2.
- Partial Fractions Followed by Binomial Expansion
- A classic 8-to-10-mark exam question gives a rational function such as f(x) = (7x - 1) / [(1 - x)(2 + x)]. First, decompose f(x) into partial fractions: A / (1 - x) + B / (2 + x). Second, rewrite each fraction with a negative index: A(1 - x)^(-1) + B(2 + x)^(-1). Third, expand each bracket up to x² or x³ (remembering to factor 2^(-1) out of the second bracket) and collect like terms.
- Numerical approximations using binomial series
- To approximate a surd or root such as √0.96 or ³√1.03 using your expansion of (1 - 2x)^(1/2), equate the expression inside the bracket to the target number (or a convenient square multiple of it): 1 - 2x = 0.96 → x = 0.02. Always check that your chosen x lies inside the range of validity |x| < 1/2, then substitute x = 0.02 into your polynomial approximation.
Quick-Fire Flashcards (6)
Click any card to check the answer, then mark it as Known to track your progress.
What is the first step when expanding (4 - 3x)^(-1/2)?
For what values of x is the binomial expansion of (4 - 3x)^(-1/2) valid?
If an expansion combines (1 + 4x)^(-1) and (2 - x)^(-1), what is the overall range of validity?
What are the simple memorisable expansions of (1 - x)^(-1) and (1 + x)^(-1) for |x| < 1?
Why would substituting x = 0.8 into the binomial expansion of (1 - 2x)^(1/2) fail to give a valid approximation?
How can you still use the expansion of (1 - 2x)^(1/2) to approximate √2 even though 1 - 2x = 2 gives x = -0.5 (outside |x| < 0.5)?
Exam-Style Questions & Mark Schemes (4)
Write or think through your answer first, then click Check Mark Scheme to see the exact points examiners award marks for.
- Q1
Find the binomial expansion of (8 - 3x)^(1/3) in ascending powers of x up to and including the term in x², simplifying each coefficient as a fraction in its lowest terms. State the set of values of x for which the expansion is valid. (5 marks)
Click “Check Mark Scheme” to compare your answer, 1. Factor 8 out of the bracket raised to the power 1/3: (8 - 3x)^(1/3) = 8^(1/3) [1 - (3/8)x]^(1/3) = 2 [1 + (-3x/8)]^(1/3) [1 mark]. 2. Apply the general binomial expansion inside the square brackets with n = 1/3 and X = -3x/8: 1 + (1/3)(-3x/8) + [(1/3)(-2/3) / 2!](-3x/8)² + . [1 mark]. 3. Simplify the terms inside the bracket: 1 - x/8 - (1/9)(9x²/64) = 1 - x/8 - x²/64 [1 mark]. 4. Multiply every term by the outside factor 2: 2 - x/4 - x²/32 [1 mark]. 5. State the validity condition: |-3x/8| < 1 → |x| < 8/3 (or -8/3 < x < 8/3) [1 mark].
- Q2
Given that f(x) = (9x + 6) / [(1 + 2x)(2 - x)²], express f(x) in partial fractions and hence find the series expansion of f(x) in ascending powers of x up to and including the term in x². State the range of values of x for which the expansion is valid. (8 marks)
Click “Check Mark Scheme” to compare your answer, 1. Set up partial fractions: (9x + 6) / [(1 + 2x)(2 - x)²] ≡ A / (1 + 2x) + B / (2 - x) + C / (2 - x)², so 9x + 6 ≡ A(2 - x)² + B(1 + 2x)(2 - x) + C(1 + 2x) [1 mark]. 2. Substitute x = 2: 24 = 5C gives C = 24/5. Substitute x = -1/2: 9(-1/2) + 6 = 3/2 = A(5/2)² = (25/4)A gives A = 6/25. Equating x² coefficients gives 0 = A - 2B, so B = 3/25 [2 marks]. 3. Rewrite in index form: f(x) = (6/25)(1 + 2x)^(-1) + (3/25)(2 - x)^(-1) + (24/5)(2 - x)^(-2) [1 mark]. 4. Expand (1 + 2x)^(-1) = 1 - 2x + 4x² - . [1 mark]. 5. Expand (2 - x)^(-1) = (1/2)(1 - x/2)^(-1) = 1/2 + x/4 + x²/8 + . and (2 - x)^(-2) = (1/4)(1 - x/2)^(-2) = (1/4)(1 + x + 3x²/4 + .) = 1/4 + x/4 + 3x²/16 + . [1 mark]. 6. Multiply by A, B, C and sum coefficients of 1, x, and x²: Constant = 6/25 + 3/50 + 6/5 = 3/2; x-term = -12/25 + 3/100 + 6/5 = 3/4 x; x²-term = 24/25 + 3/200 + 9/10 = 375/200 x² = 15/8 x²; so f(x) ≈ 3/2 + (3/4)x + (15/8)x² [1 mark]. 7. Validity: |2x| < 1 → |x| < 1/2 and |x/2| < 1 → |x| < 2; taking the stricter intersection gives |x| < 1/2 [1 mark].
- Q3
Find the first three terms in the binomial expansion of √((1 + 2x) / (1 - 2x)) in ascending powers of x, and state the range of values of x for which the expansion is valid. (5 marks)
Click “Check Mark Scheme” to compare your answer, 1. Rewrite the square root quotient using fractional and negative indices: √((1 + 2x) / (1 - 2x)) = (1 + 2x)^(1/2) (1 - 2x)^(-1/2) [1 mark]. 2. Expand (1 + 2x)^(1/2) up to x²: 1 + (1/2)(2x) + [(1/2)(-1/2) / 2](2x)² = 1 + x - (1/2)x² + . [1 mark]. 3. Expand (1 - 2x)^(-1/2) up to x²: 1 + (-1/2)(-2x) + [(-1/2)(-3/2) / 2](-2x)² = 1 + x + (3/2)x² + . [1 mark]. 4. Multiply the two expansions together and retain terms up to x²: (1 + x - 1/2 x²)(1 + x + 3/2 x²) = 1(1 + x + 3/2 x²) + x(1 + x) - 1/2 x²(1) = 1 + 2x + 2x² [1 mark]. 5. State the validity condition: |2x| < 1 → |x| < 1/2 [1 mark].
- Q4
The binomial expansion of (1 + ax)^n, where n < 0, begins with 1 - 6x + 24x² + kx³ + . Find the values of the constants a, n, and k. (5 marks)
Click “Check Mark Scheme” to compare your answer, 1. Write out the general binomial expansion of (1 + ax)^n: 1 + n(ax) + [n(n - 1) / 2]a²x² + [n(n - 1)(n - 2) / 6]a³x³ + . [1 mark]. 2. Equate coefficients of x and x²: na = -6 (so a = -6 / n) and [n(n - 1) / 2]a² = 24 [1 mark]. 3. Substitute a = -6 / n into the x² equation: [n(n - 1) / 2](36 / n²) = 24 → 18(n - 1) / n = 24 [1 mark]. 4. Solve for n and a: 18n - 18 = 24n → 6n = -18 → n = -3; therefore a = -6 / (-3) = 2 [1 mark]. 5. Calculate k, the coefficient of x³: k = [(-3)(-4)(-5) / 6] × 2³ = (-10) × 8 = -80 [1 mark].
Common Exam Questions (4)
What is the #1 algebraic slip in Year 2 Binomial Expansion questions?
Forgetting to raise the factored-out constant a to the power n (writing 4(1 + x/4)^(1/2) instead of 4^(1/2)(1 + x/4)^(1/2) = 2(1 + x/4)^(1/2)), or forgetting to put brackets around the whole x-term when squaring (writing -3x² instead of (-3x)² = +9x²).
How do I expand a fraction with a numerator and denominator like (2 + 3x) / (1 - x)²?
Rewrite it as a product: (2 + 3x)(1 - x)^(-2). First expand (1 - x)^(-2) up to x² or x³, then multiply the resulting polynomial by (2 + 3x) and collect like terms up to the required power.
Can |x| = 1 ever be included in the validity interval for a negative or fractional power?
At A-Level, the standard formula book states |x| < 1 (strict inequality), and mark schemes always expect strict inequality (|x| < a/b or -a/b < x < a/b) without equals signs.
How can I verify my binomial expansion on my calculator during an exam?
Pick a tiny number inside the validity range, such as x = 0.001, and plug it into both the original function (e.g. (8 - 3(0.001))^(1/3)) and your quadratic/cubic expansion (2 - 0.001/4 - 0.001²/32). They should agree to at least 7 or 8 decimal places.
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