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Expanding (a + bx) to a negative or fractional power

Factor the constant before expanding (a + bx) to negative or fractional powers. Exact coefficients, roots, reciprocals, validity and worked practice.

Before you startGeneralised binomial expansion and index laws

01 / Make the bracket start with 1

The outside constant also carries the exponent.

The standard expansion starts with (1 + z)ᵖ. If the constant inside your bracket is not 1, factor it out first. Then raise that outside factor to p and keep it through every line.

Use the model to follow four expressions one step at a time. The final stage distributes the outside factor into every coefficient.

Keep track of the outside factorExplore

√(9 + 6x)

1 · Identify the constant. At x = 0, the answer must be 3.

√(9 + 6x)

The whole bracket is raised to ½. Pulling out 9 will contribute 9^(½), not 9.

Guaranteed expansion interval: |x| < 3/2.

Expression Working stage 1 of 4

Move the stage control yourself. Every line also appears in the worked notes; the model never advances automatically.

Watch the constant factor travel through the expansion

Pause, replay or seek freely. The notes explain the same idea and stay in view.

02 / The general method

Identify the outside multiplier and the scaled input separately.

(a + bx)ᵖ = aᵖ(1 + bx/a)ᵖ, for a > 0
Expand in z = bx/a, then multiply every term by aᵖ.

For a negative or non-integer exponent, use the guaranteed interval |bx/a| < 1. If b ≠ 0 this means |x| < a/|b| when a > 0. Within it, 1 + bx/a is positive, so real fractional powers are unambiguous.

Through degree 3:
aᵖ[1 + p(b/a)x + p(p − 1)(b/a)²x²/2
+ p(p − 1)(p − 2)(b/a)³x³/6 + …]

Check the constant term against the original expression at x = 0: it must be aᵖ. This check detects a missing outside factor. It does not verify every later coefficient.

03 / A square-root example

Take the square root of the factored constant.

Expand √(9 + 6x) through x³Worked example

√[9(1 + 2x/3)] = 3(1 + 2x/3)^(½)

The outside multiplier is √9 = 3; the new input is z = 2x/3.

3[1 + ½(2x/3) − ⅛(2x/3)² + (1/16)(2x/3)³ + …]

Apply the standard square-root coefficients inside the bracket.

= 3[1 + x/3 − x²/18 + x³/54 + …]

Simplify inside before distributing.

= 3 + x − x²/6 + x³/18 + …

Multiply every term by 3.

Guaranteed for |x| < 3/2

Solve |2x/3| < 1, not |6x| < 1.

The function itself is real for x ≥ −3/2, but that is a different question from the guaranteed convergence interval of this expansion about zero.

04 / A reciprocal example

The outside multiplier is the reciprocal of the constant.

Expand 1/(4 − 2x) through x³Worked example

(4 − 2x)⁻¹ = ¼(1 − x/2)⁻¹

Factoring out 4 contributes 4⁻¹ = ¼.

= ¼[1 + x/2 + x²/4 + x³/8 + …]

This is a geometric expansion with ratio x/2.

= ¼ + x/8 + x²/16 + x³/32 + …

The factor ¼ multiplies all coefficients.

Guaranteed for |x| < 2

The transformed input is −x/2.

A quick check: setting x = 0 in either expression gives ¼. A constant term of 4 or 1 would reveal a factoring error.

05 / A negative fractional power

Evaluate the constant using roots and reciprocals.

Expand (8 + 4x)^(−⅔) through x³Worked example

= 8^(−⅔)(1 + x/2)^(−⅔) = ¼(1 + x/2)^(−⅔)

The cube root of 8 is 2; its inverse square is ¼.

Coefficients for p = −⅔: 1, −⅔, 5/9, −40/81

Use the falling products, including their signs.

= ¼[1 − x/3 + 5x²/36 − 5x³/81 + …]

Substitute z = x/2 and raise the entire fraction to each power.

= ¼ − x/12 + 5x²/144 − 5x³/324 + …

Distribute ¼.

Guaranteed for |x| < 2

The positive-bracket interval also keeps the real fractional-power calculation safe.

A cube-root expression may have a wider real domain. That does not extend this expansion’s guaranteed interval about zero.

06 / Keep surd coefficients exact

A factored exact answer can be clearer than a decimal expansion.

Expand √(12 + 3x) through x³Worked example

= 2√3(1 + x/4)^(½)

√12 = 2√3, not 6 or 12.

= 2√3[1 + x/8 − x²/128 + x³/1024 + …]

Substitute x/4 into the standard coefficients.

= 2√3 + (√3/4)x − (√3/64)x² + (√3/512)x³ + …

This distributed form has the same exact coefficients.

Guaranteed for |x| < 4

The outside factor does not alter the input condition.

Either exact form is useful if the question does not demand a particular layout. Avoid rounding √3 before multiplying coefficients.

07 / When the constant needs extra care

Do not apply real fractional-power identities blindly to negative bases.

The general real-power method above assumes a > 0. For negative a and a non-integer exponent, first check whether the original expression is real near x = 0 and whether the particular power identity is valid. For example √(−4 + x) is not real in any neighbourhood of zero, so it has no real binomial expansion there.

An integer power with a negative constant: (−2 + x)⁻²Worked example

= (−2)⁻²(1 − x/2)⁻² = ¼(1 − x/2)⁻²

Integer powers make this factorisation valid where the denominator is nonzero.

= ¼[1 + x + 3x²/4 + x³/2 + …]

Use p = −2 and z = −x/2.

= ¼ + x/4 + 3x²/16 + x³/8 + …, for |x| < 2

The original denominator also excludes x = 2.

If a = 0 you cannot divide by a: simplify (bx)ᵖ directly instead. If b = 0 the expression is constant wherever defined. A nonnegative integer p gives a finite polynomial, with no infinite-series convergence restriction.

08 / Diagnose missing factors

Check both the constant and the transformed input.

A learner writes √(25 − 10x) = 5 − x − x²/10 − x³/50Worked example

Factoring gives 5(1 − 2x/5)^(½)

The displayed four coefficients are correct.

The written equality is incomplete for general x

Use an ellipsis for the infinite expansion, or ≈ for the four-term approximation.

Can √(−9 + 3x) be expanded as a real binomial series about x = 0? Explain.

Hint

Determine where the square root is real.

Worked solution

No. It is real only for x ≥ 3, so it is not real in a neighbourhood of zero. Factoring out −9 and applying the ordinary real square-root expansion is invalid.

10 / Recap

Normalise, expand, distribute, then check.

Factor the constant so the bracket starts with 1.
Raise the outside constant to the original exponent.
Expand using the complete input bx/a.
Multiply every coefficient by the outside factor.
Keep exact fractions and surds.
Check x = 0, the real domain and the guaranteed interval.

Review the coefficient formula →

Section 1 of 10 · Make the bracket start with 1