Multiply It Back — The 10-Second Check That Catches Almost Every Factoring Mistake

You factored x^2 + x - 6 as (x + 3)(x - 2). Are you sure? You did the work, you got something — but the sign on the middle term in your bracket and the sign in the original have a habit of crossing wires. The fastest way to know is not to redo the factoring. It is to multiply your answer back out and check it lands on the original polynomial.

This is the single most undervalued habit in school algebra. Treat factoring as a closed loop: the moment you write down the factored form, expand it back. If it matches, sign and move on. If it does not, you know immediately you have a sign slip or a missing factor — before the question is marked, before the next part of the problem propagates the error, before you waste ten more minutes on a wrong base.

The cost is about ten seconds. The saving is everything from one careless mark to a whole question.

Why this works — factoring is division

When you factor x^2 + x - 6 into (x + 3)(x - 2), what you have actually done is rewrite the polynomial as a product. The reverse operation — multiplying (x + 3)(x - 2) — must give you back exactly x^2 + x - 6. No new terms, no missing terms, every sign in place.

Why this is reliable: factoring is the inverse of multiplication, the same way division is the inverse of multiplication for numbers. If 12 \div 3 = 4, you can check by multiplying 3 \times 4 = 12. If x^2 + x - 6 = (x + 3)(x - 2), you can check by multiplying (x + 3)(x - 2) back out. The forward operation (factoring) is hard — guesswork, pattern recognition, sign tracking. The inverse operation (multiplying) is mechanical — distribute, collect, done. So you let the easy direction police the hard one.

The whole loop fits on a postcard.

How to multiply back in your head

For a quadratic that factored into two binomials, FOIL is the standard mnemonic — First, Outer, Inner, Last. For (x + 3)(x - 2):

• F (first × first): x \cdot x = x^2.
• O (outer × outer): x \cdot (-2) = -2x.
• I (inner × inner): 3 \cdot x = 3x.
• L (last × last): 3 \cdot (-2) = -6.

Add them: x^2 - 2x + 3x - 6 = x^2 + x - 6. Why FOIL: the distributive law says you multiply every term in the first bracket by every term in the second. With two terms each, that gives four products. FOIL is just a fixed order so you do not forget any.

For three or more factors, expand two at a time, or use the box method (a small grid where rows and columns are the bracketed terms and each cell holds the product). The box method is harder to skip a term in, because every cell is visible.

Worked example 1 — clean match, ship it

You are asked to factor x^2 + 7x + 12. You spot the pair 3, 4 (multiply to 12, add to 7) and write:

Worked example 2 — the sign-slip rescue

You are factoring x^2 + x - 6. You think for a moment, jot down two numbers that multiply to -6 and add to +1, and write:

It looks right. Both signs are in there, the numbers 3 and 2 multiply to 6. But you developed the habit, so before moving on you multiply back.

This is the most common single mistake students make in factoring — the sign of the middle term in (x \pm a)(x \pm b) when a and b have opposite signs. The verify-by-multiplying-back check catches it every single time, because it is impossible to multiply (x - 3)(x + 2) and not notice the middle term comes out to -x.

Worked example 3 — the dropped-factor rescue

You are asked to factor 2x^2 + 4x. You spot the common factor x (you forget the 2) and write:

You always pull the GCF first, but in the rush you grabbed only part of it. Multiply back to check.

This kind of slip — pulling out a partial GCF — is invisible to your eye when you stare at the original. But the moment you multiply back, the ratio of every coefficient to its original is exactly the missing factor, and you spot it instantly.

What the check catches, in order of frequency

After grading enough Class 9 papers, the most common factoring slips fall into four buckets — and the multiply-back check catches all four:

1. Sign of the middle term wrong (x + 1 instead of x - 1 in one bracket). The product's middle coefficient flips sign or magnitude. Always shows up.
2. Forgot a constant factor (x(x+2) for 2x^2 + 4x). Every term in your product is off by the same multiplicative ratio.
3. Forgot a variable factor (treating x^3 - x as (x-1)(x+1) instead of x(x-1)(x+1)). Your product has the wrong degree.
4. Wrong split of the middle term in the ac-method, giving brackets that multiply to a quadratic with the wrong middle coefficient. Caught at FOIL.

The unifying observation is that when factoring is wrong, the wrongness always surfaces in the product. There is no slip that hides from a careful redistribution. So if you make checking automatic, you make wrong factorisations into a vanishing problem.

The cricket-net analogy

Think of it like batting in the nets. After every shot, you look at where the ball went — did it travel along the ground in front of square, or did it pop up off the edge? The look-back takes a fraction of a second and tells you whether the shot was clean. You do not skip it because then you never know which shots are working.

Multiplying back after factoring is the same look-back. You hit the shot (factored), now check where the ball went (multiplied back). Match? Clean shot. Mismatch? Adjust your grip, try again. Over a few hundred reps the loop becomes invisible — you are factoring and verifying in one motion, the way a good batter plays and looks up at the same instant.

Common confusions

• "If the answer is in the textbook, I do not need to check." Wrong — your handwriting is what gets marked, not the textbook. A miscopied sign in your script is still a wrong answer. Verify your written form, not the printed solution.

• "FOIL is too slow for the exam." FOIL on a binomial product is four products and a sum — about eight seconds. The "saving" you make by skipping is exactly the time you would have spent verifying, except now you do not know whether your answer is right.

• "I'll check at the end with all my answers." No, check inline, immediately. If your factoring is wrong and the next step uses it (substituting into the quadratic formula, dividing, or simplifying a rational expression), you propagate the error and the next several lines are also wrong. Verify at the line where the factoring happens.

• "The sign in the middle bracket is the same as the sign in the trinomial — so checking is pointless." Only when both numbers in the brackets have the same sign. When they have opposite signs (the negative-constant case like x^2 + x - 6), the bracket signs are the most error-prone single decision in factoring. That is exactly where the verify habit pays off most.

The bottom line

Every factorisation has a built-in check: multiply it back. The check takes ten seconds, costs nothing, and catches almost every common error — sign slips, dropped factors, miscounted terms. Make it a reflex. The moment you write the factored form, your hand should already be moving to FOIL the brackets back into a polynomial.

Factoring is division; multiplying back is the inverse check. Run the inverse every single time, and you stop submitting wrong factorisations.

References

1. FOIL method — Wikipedia
2. Factorization — Wikipedia
3. Khan Academy — Multiplying binomials
4. NCERT Class 9 Maths, Chapter 2 — Polynomials
5. Distributive property — Wikipedia