Jon P. answered • 07/05/15
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Since there are two variables, a and b, it is a function of both.
The way to write that is f(a, b) = a4 - 7a²b² + kb4.
Daniel K.
For this how should i do it?
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07/05/15
Jon P.
tutor
Oh, I see why you wanted to know how to express the polynomial as a function of on of the variables.
To use the remainder theorem for this, you're best to consider it a function of a. So the remainder when dividing by a-3b will be f(3b). Let's see what happens:
f(a) = a4 - 7a²b² + kb4
f(3b) = (3b)4 - 7(3b)2b2 + kb4 =
34b4 - 7 (32)b4 + kb4 =
81b4 - 63b4 + kb4 =
18b4 + kb4 = (18 + k)b4
You want the remainder to be 0 for all b, so that means k would have to be -18 and the polynomial would be a4 - 7a²b² - 18b4.
You can factor this into (a2 - 9b2)(a2 + 2b2) = (a - 3b)(a + 3b)(a2 + 2b2)
And it does turn out that a - 3b is a factor, which means that the remainder when dividing by a - 3b is 0, as expected.
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07/05/15
Daniel K.
you said Oh, I see why you wanted to know how to express the polynomial as a function of on of the variables. What do u mean by a function of on of the variables?
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07/05/15
Jon P.
tutor
Sorry! That was a typo. I meant to say, "I see why you wanted to know how to express the polynomial as a function of ONE of the variables."
My point was that in order to use the remainder theorem, you have to think of the polynomial as a function of a specific variable, and use that to find the remainder. When you first asked the question, it just looked like you wanted to understand how to write a function of more than one variable. But when you explained the actual problem, it was clear that you were looking for something different.
I hope that clarifies things.
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07/06/15
Daniel K.
thanks
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07/06/15
Daniel K.
07/05/15