I’m revising for a test and expanding brackets is driving me a bit bananas. I keep tripping over the negatives and coefficients, like when there’s a minus outside or when I’m multiplying two brackets together. It feels like unpacking groceries and realizing I left a bag in the car-something always gets missed or the sign flips on me! How do you reliably expand things like (x − a)(x + b) or -2(3x − 4) without messing up the signs? Is there a simple rule-of-thumb or mental checklist that you use every time? Also, when there are three factors like (x + 1)(x − 2)(x + 3), should I always expand two first and then multiply the result, or is there a quicker test-friendly trick to keep it clean and accurate? I’m preparing for a test soon and would love a clear way to avoid those sign mistakes.
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3 Responses
My rule-of-thumb is: carry the sign with the term, and rewrite subtraction as addition of a negative, e.g. x − a becomes x + (−a). Then distribute one term at a time and decide the sign by counting negatives in that product (odd number ⇒ negative, even ⇒ positive). For example, −2(3x − 4) = (−2)(3x) + (−2)(−4) = −6x + 8. For (x − a)(x + b) = (x + (−a))(x + b), multiply all pairs: x·x = x^2, x·b = +bx, (−a)·x = −ax, (−a)·b = −ab, so x^2 + (b − a)x − ab. Two quick checks keep signs honest: the constant term must be the product of the constants (here (−a)·b = −ab), and the x-coefficient must be their sum (here b + (−a) = b − a). With three factors, expand two first (pick the easiest pair), then multiply by the third; a clean check is the pattern (x + p)(x + q)(x + r) = x^3 + (p + q + r)x^2 + (pq + pr + qr)x + pqr, so (x + 1)(x − 2)(x + 3) should be x^3 + 2x^2 − 5x − 6. If you’re prone to slips, write every intermediate term before combining like terms and, if needed, line them up by powers of x. I might be slightly over-careful here, but these habits make the sign errors disappear.
Golden rule: treat a leading minus as “multiply by −1”-distribute signs first, then numbers-and use (x+r)(x+s)=x^2+(r+s)x+rs, so (x−a)(x+b)=x^2+(b−a)x−ab and −2(3x−4)=−(6x−8)=−6x+8. For three factors, expand the pair with a nice r+s first (e.g., (x+3)(x−2)=x^2+x−6), then multiply by the third and sanity-check that the constant equals the product of constants-want to try a quick one and say which pairing felt cleanest?
My sticky-note rule: glue each term to its sign and distribute like a careful grocer-decide the sign first (−×+=−, −×−=+), so (x−a)(x+b)=x^2+bx−ax−ab and −2(3x−4)=−6x+8; for three factors, multiply any convenient pair first (hunt for friends like (x+a)(x−a)) and then distribute again. Want a visual safety net like the box method to catch sneaky negatives? Try this: https://www.khanacademy.org/math/algebra/polynomial-factorization/expanding-expressions/v/multiplying-binomials – which pair would you expand first in (x+1)(x−2)(x+3)?