How to remember the signs in a³ − b³ = (a−b)(a² + ab + b²): SOAP for the middle term

In short

The identity a^3 - b^3 = (a-b)(a^2 + ab + b^2) has three signs you have to get right, and the middle one — the +ab — is the one that almost everyone fumbles. The fix is one word: SOAP, which stands for Same, Opposite, Always Positive. The linear bracket takes the same sign as the original (-b in the cubes gives -b in the bracket). The middle of the quadratic takes the opposite sign (so the original minus becomes a plus). The last term (b^2) is always positive, because it is a square. Three letters, three signs, and the formula stops being a memory test.

You have seen it on the worksheet: a^3 - b^3 = (a - b)(a^2 + ab + b^2). The first bracket feels obvious — it is just a minus b. The last term in the second bracket also feels obvious — b^2 is a square, of course it is positive. The thing that derails everyone is the middle. Why is it +ab? You started with a^3 minus b^3. Where did the plus come from? And how do you know it is not -ab?

This article is about that one sign. By the end you will have a one-word mnemonic locked in (SOAP), a picture of why each letter is forced, and an algebraic check that proves the formula could not work any other way.

SOAP — three letters for three signs

Look carefully at the identity. There are three signs hidden inside it.

a^3 \underbrace{-}_{\text{1}} b^3 = (a \underbrace{-}_{\text{2}} b)(a^2 \underbrace{+}_{\text{3}} ab + b^2)

Sign 1 is the original — the minus between a^3 and b^3. That one is given to you by the problem. It is the anchor. Signs 2 and 3 are the ones you have to produce. SOAP tells you exactly what to do.

S — Same. The sign in the linear bracket (sign 2) is the same as the anchor. Anchor is -, so the linear bracket is (a - b). If the anchor were + (i.e. a^3 + b^3), the linear bracket would be (a + b).
O — Opposite. The middle sign of the quadratic bracket (sign 3) is the opposite of the anchor. Anchor is -, so the middle is +ab. If the anchor were +, the middle would be -ab.
A and P — Always Positive. The final b^2 in the quadratic does not depend on the anchor at all. It is always +b^2. Why: it is a square. Squares of real numbers are non-negative, and the identity has to hold for every real a and b, including the case where a and b have opposite signs. Putting a minus in front of b^2 would break the identity instantly — try a = 1, b = 1: a^3 - b^3 = 0, but (a-b)(a^2 + ab - b^2) = 0 \cdot 1 = 0 by accident, while a = 2, b = 1 gives 7 from the cubes and (1)(4 + 2 - 1) = 5 from the wrong formula. Mismatch. The +b^2 is forced.

Three letters. Three signs. Done.

Both cube identities side by side, with the three signs colour-coded. The red signs (linear bracket) match the anchor. The blue signs (middle of quadratic) flip. The green sign (final term) never moves. Read SOAP off the picture once and you have it forever.

Three worked examples

Example 1 — the difference: $a^3 - b^3$

Anchor sign: -.

Same: linear bracket is (a - b). Carries the anchor.
Opposite: middle of quadratic is +ab. Flips the anchor.
AP: final term is +b^2. Always.

Putting it together:

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Spot-check with a = 3, b = 2: left side is 27 - 8 = 19. Right side is (3-2)(9 + 6 + 4) = 1 \times 19 = 19. Match.

Example 2 — the sum: $a^3 + b^3$

Anchor sign: +.

Same: linear bracket is (a + b).
Opposite: middle of quadratic is -ab (anchor was +, so flip to -).
AP: final term is +b^2. Same as before — always.

Putting it together:

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Spot-check with a = 3, b = 2: left side is 27 + 8 = 35. Right side is (3+2)(9 - 6 + 4) = 5 \times 7 = 35. Match.

Example 3 — verify $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$ by expansion

Watch every cross-term cancel. Distribute a across the second bracket, then -b:

\begin{aligned} (a-b)(a^2 + ab + b^2) &= a \cdot a^2 + a \cdot ab + a \cdot b^2 \\ &\quad + (-b) \cdot a^2 + (-b) \cdot ab + (-b) \cdot b^2 \\ &= a^3 + a^2 b + ab^2 - a^2 b - ab^2 - b^3 \end{aligned}

Now group the cancellation pairs:

a^3 \underbrace{+\, a^2 b - a^2 b}_{= \ 0} \underbrace{+\, ab^2 - ab^2}_{= \ 0} - b^3 = a^3 - b^3 \checkmark

Both cross-pairs vanish. Why does this only work with +ab in the middle? When you distribute a over +ab you get +a^2 b. When you distribute -b over a^2 you get -a^2 b. Those two have opposite signs, so they cancel. Same story for the ab^2 pair: a \cdot b^2 = +ab^2 and -b \cdot ab = -ab^2, opposite signs, cancel. If the middle had been -ab instead, then a \cdot (-ab) = -a^2 b would match (not cancel) the -a^2 b from -b \cdot a^2, and you would end up with -2a^2 b left over. The whole thing would no longer simplify to a^3 - b^3. The +ab in the middle is not a stylistic choice — it is the unique sign that makes the cross-terms cancel.

A common confusion: "but the original was minus, why does the middle become plus?"

Students see a^3 - b^3 and instinctively put a minus everywhere. They write (a - b)(a^2 - ab + b^2), which is wrong. The instinct is understandable — minus in, minus out — but the algebra demands the opposite, and SOAP encodes exactly that.

Here is the intuition. The product (a - b)(\text{quadratic}) has to equal a difference of cubes. That means most of the quadratic's job is to make all the cross-terms disappear, leaving only a^3 and -b^3. For the cross-terms to cancel, the middle of the quadratic has to push back against the sign of the linear bracket. The linear bracket says minus, so the middle says plus. They are doing opposite work. That is the O in SOAP.

If you ever get stuck mid-problem, run the cancellation logic in your head: "Linear bracket is minus, so to cancel cross-terms the middle must be plus." That sentence is SOAP unfolded.

Quick recognition drill

Run through these and check that you produce the right factorisation each time. The anchor sign is in bold.

x^3 \mathbf{-} 1 = (x - 1)(x^2 + x + 1) — anchor -, so middle is +x.
x^3 \mathbf{+} 1 = (x + 1)(x^2 - x + 1) — anchor +, so middle is -x.
8 \mathbf{-} y^3 = (2 - y)(4 + 2y + y^2) — anchor -, so middle is +2y.
27 \mathbf{+} y^3 = (3 + y)(9 - 3y + y^2) — anchor +, so middle is -3y.
125 \mathbf{-} 8x^3 = (5 - 2x)(25 + 10x + 4x^2) — anchor -, so middle is +10x.

Every one follows the same three-step recipe. Anchor → S, O, AP. No guessing, no flipping back to check the textbook, no second-guessing in the exam.

The cricket-bat take

Imagine a cricket bat carved out of a cube of willow a cm on a side, with a smaller cube of waste wood b cm on a side trimmed off. The volume of usable bat is a^3 - b^3. SOAP tells you instantly that this volume splits as

(a - b) \cdot (a^2 + ab + b^2)

— a flat slab of "thickness" (a-b) and cross-section (a^2 + ab + b^2). The cross-section is bigger than a^2 because of that +ab + b^2, which is exactly the geometric correction for the bookkeeping you did when you nested the two cubes inside the same big cube. (For the full 3D picture, see the geometric proof of the sum-of-cubes identity — the difference case is its mirror image.)

The takeaway

The middle sign in a^3 - b^3 = (a - b)(a^2 + ab + b^2) is positive because the cancellation of cross-terms during expansion requires it. SOAP — Same, Opposite, Always Positive — turns that algebraic necessity into a one-word recipe you can run in three seconds. Same anchor in the linear bracket. Opposite of the anchor in the middle. Always Positive at the end.

Three letters. Three signs. One identity you will never get wrong again.

References

Algebraic identities — the parent article with all seven standard identities.
Sum and difference of cubes: SOAP mnemonic — sibling article focused on the broader sum/difference pair.
Geometric proof of a^3 + b^3 — the 3D dissection that makes the \mp ab middle term inevitable.
Polynomial factorization — where SOAP sits in the bigger toolkit.
NCERT Class 9 Mathematics, Chapter 2: Polynomials — the textbook treatment of a^3 \pm b^3.
Wikipedia: Sum and difference of cubes — algebraic statement and history.