I’m revising fundamentals and trying to be consistent with expanding brackets, especially when there’s a minus in front of something big.
Example I’m working on: 4(3a – 2b) – (a – b)(a + 2b).
My attempt:
– 4(3a – 2b) -> 12a – 8b
– (a – b)(a + 2b) -> a^2 + ab – 2b^2
– Then subtracting the second part: 12a – 8b – a^2 + ab + 2b^2
I’m not confident about the sign on the ab term after the subtraction – I think that’s where I’m slipping. Could someone point out exactly how the signs should change when there’s a minus in front of a whole product like this? Also, any quick check to avoid dropping or flipping a term would help. I’m just trying to tighten up the basics.
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3 Responses
Think of the leading minus as “multiply by −1”-it flips every sign of the entire expanded polynomial: (a−b)(a+2b)=a^2+ab−2b^2, so 4(3a−2b)−(a−b)(a+2b)=12a−8b−a^2−ab+2b^2.
Quick check vibe: −(x−2y+3)=−x+2y−3 (every sign inside switches).
You’ve got the product right, the sign flip is the only slip. Do it in two clean steps and keep the brackets until the very end: 4(3a − 2b) = 12a − 8b, and (a − b)(a + 2b) = a^2 + ab − 2b^2. Now subtract the whole product, which means multiply it by −1 and flip every sign: 12a − 8b − (a^2 + ab − 2b^2) = 12a − 8b − a^2 − ab + 2b^2. So the ab term should be negative. A quick pattern check helps: inside you had +, +, −; after the leading minus you must see −, −, +. If you don’t, something went walkabout.
When I kept botching these, I forced myself to rewrite the minus as a factor: −(a − b)(a + 2b) = (−a + b)(a + 2b). Pushing the minus into one factor first made the multiplication routine, no surprises at the end. Another fast sanity check: plug easy numbers. With a = b = 1, the original gives 4(1) − 0 = 4. The correct expansion 12a − 8b − a^2 − ab + 2b^2 gives 12 − 8 − 1 − 1 + 2 = 4. If you accidentally keep +ab, you get 6-instant red flag. Keep the brackets until you flip all the signs, or shove the minus into one factor before you multiply. Either way, no dropped signs.
I’d treat the leading minus as “multiply by −1”: expand (a−b)(a+2b)=a^2+ab−2b^2, then flip all the signs to get −a^2−ab+2b^2, so 4(3a−2b)−(a−b)(a+2b)=−a^2−ab+12a−8b+2b^2. If you’re wobbly, plug a=b=1-your +ab version gives 6, while this one (I’m pretty sure!) gives 4.