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Find the remainder of f(x) divided by x^2 - 1 when f(x) = x^10 - 8x^8+7x^4+ 8x^3-12x^2 -15x - 5.

 Aug 6, 2022
 #1
avatar+113 
+1

As before, use the onilne calculator, here is the answer:

https://www.emathhelp.net/calculators/algebra-1/polynomial-long-division-calculator/?numer=x%5E10+-+8x%5E8%2B7x%5E4%2B+8x%5E3-12x%5E2+-15x+-+5&denom=+x%5E2+-+1

And an image of the solution:

The fractional expression is the last term in this, so the remainder is -7x -17.

 Aug 6, 2022
edited by tuffla2022  Aug 6, 2022
 #2
avatar+252 
+4

No, there is no need to use a calculator for this problem.

 

f(x) = (x^2 - 1)q(x) + r(x)

 

We know that r(x) is a linear equation in the form of ax + b. 

 

f(1) = -24 = a + b (The first term cancels out)

f(-1) = -10 = -a + b

 

Therefore, we get two system of equations.

 

a + b = -24

-a + b = -10

 

b = -17

a = -7

 

Hence, the remainder is -7x - 17. 

 Aug 6, 2022
 #3
avatar+113 
+2

Well, seems like we agree on the answer, but I haven't seen Your cool technique before. You learn something new every day.

tuffla2022  Aug 6, 2022

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