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Algebra/Multiplying negative and positive variables...


QUESTION: I'm having trouble with a question in my algebra self-study course. The question is written out as follows:

-k^2(ak^3 - 3 + a)

I'm being asked to multiply a monomial by the polynomial in parentheses using the distributive property, which I have no trouble with until I reach the last term of the polynomial. I'm having trouble with -k^2(a)

The answer I gave was -k^2 a ...but the answer given by my textbook was -ak^2.

I understand that I'm being asked to multiply the negative variable (-k^2) by the positive variable (a), and that normally this would result in a negative answer just like any time you multiply a positive value by a negative value...but I'm under the impression that (-ak^2) is the equivalent of writing out (-a)(k^2), where the k is positive...the negative sign applies only to the literal variable/factor immediately following it right? I'm just confused because if that is the case then my textbook is saying that (-k^2)(a) and (-ak^2) are equivalent, when the variables have switched signs, and if given definite values produce different answers.

That's what's confusing me - how can, for example (-b^2)(a) be equivalent to (-ab^2)?

If a=3 and b=2, the negativity or positivity of the number being raised to the exponent will change the final product for each expression, making them not equivalent...?

I'm not quite sure where I'm going wrong!

ANSWER: Your answer -k^2a makes it look like 2a is the exponent whereas a multiplies the quantity -k^2. (-k^2)(a) is equivalent to (-ak^2).  In your example, b^2 is 9 in both cases.  If it had been (-b)^2 then the answer would have been 12 instead of -12. You use parentheses for clarification.  

---------- FOLLOW-UP ----------

QUESTION: Thanks for the quick reply Richard!

I'm still not quite getting it. I guess what is confusing me is I still don't see, going with my "b" example, how the values are equivalent...I understand that b^2 and -b^2 give the same result (because any negative squared yields a positive). In (-b^2)(a) you've essentially got (-2^2)(3) = 12, but then in (-ab^2), which we're saying is equivalent, this time the 3 [the "a"] is signed as negative instead of the power of 2, coming out as (-3)(2^2) which yields a product of -12.

So if (-b^2)(a) evaluates to (-2^2)(3) = 12 ...
and if (-ab^2) evaluates to (-3)(2^2) = -12 ... can (-b^2)(a) and (-ab^2) be said to be equivalent? I just can't understand why "a" becomes a negative value, or how the two are equal to one another.

Thanks again for your help, I appreciate it!

No, b^2 and -b^2 are not the same.  with (-b^2)(a) you're not squaring -b, as I indicated previously, that would be (-b)^2
In your last statement, a does not become a negative value.  It's like -(a)b^2 so the whole expression is negative.   (-b^2)(a) evaluates to (-4)(3) = -12
Does that settle it?  


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