# Thread: Question about Multiplying a constant and two polynomials

1. ## Question about Multiplying a constant and two polynomials

Lets say I have 9(2x^2-4x+5)^8(4x-4). Does the 9 get distributed to both polynomials? I'm attempting to differentiate using the product and extended power rule from calculus, but need to understand the Algebra behind it all. Any pointers would be greatly appreciated.

2. ## Re: Question about Multiplying a constant and two polynomials

Originally Posted by shane805
Lets say I have 9(2x^2-4x+5)^8(4x-4). Does the 9 get distributed to both polynomials? I'm attempting to differentiate using the product and extended power rule from calculus, but need to understand the Algebra behind it all. Any pointers would be greatly appreciated.
Shane

You need to remember the basic idea behind elementary algebra: letters and expressions just represent numbers.

$5 * 2^2 * 3 = 5 * 4 * 3 = 60 \ne (5 * 2^2) * (5 * 3) = 20 * 15 = 300.$

However, I think I know what is bothering you with respect to the differentiation. This problem requires you to use the chain rule (at least implicitly) in addition to the product and power rules. In my opinion, until you hum at differentiating, it is best to use the chain rule explicitly.

$y = 9(2x^2 - 4x + 5)^8(4x - 4)\ and\ w = (2x^2 - 4x + 5)^8(4x - 4) \implies$

$y = 9w \implies \dfrac{dy}{dw} = 9 \implies \dfrac{dy}{dx} = \dfrac{dy}{dw} * \dfrac{dw}{dx} = 9 * \dfrac{dw}{dx}.$

Now you know what to do with the 9. All you have left to do is to deal with $(2x^2 - 4x + 5)^8(4x - 4).$

Personally I'd recommend using the chain rule explicitly some more if you have any uncertainty. Hope this helps. If not, ask a follow up question.

3. ## Re: Question about Multiplying a constant and two polynomials

Originally Posted by JeffM
You need to remember the basic idea behind elementary algebra: letters and expressions just represent numbers.
$5 * 2^2 * 3 = 5 * 4 * 3 = 60 \ne (5 * 2^2) * (5 * 3) = 20 * 15 = 300.$
However, I think I know what is bothering you with respect to the differentiation. This problem requires you to use the chain rule (at least implicitly) in addition to the product and power rules. In my opinion, until you hum at differentiating, it is best to use the chain rule explicitly.
$y = 9(2x^2 - 4x + 5)^8(4x - 4)\ and\ w = (2x^2 - 4x + 5)^8(4x - 4) \implies$
$y = 9w \implies \dfrac{dy}{dw} = 9 \implies \dfrac{dy}{dx} = \dfrac{dy}{dw} * \dfrac{dw}{dx} = 9 * \dfrac{dw}{dx}.$
Now you know what to do with the 9. All you have left to do is to deal with $(2x^2 - 4x + 5)^8(4x - 4).$
Personally I'd recommend using the chain rule explicitly some more if you have any uncertainty. Hope this helps. If not, ask a follow up question.
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