Polynomial Division - Finding The Polynomial and Finding The Other Factors?

Question

When a certain polynomial is divided by x + 2, the quotient is x^2 -4x + 1 and the remainder is 8. What is the polynomial?

Now the division of polynomials is a simple idea, but the learning annex that supplies these courses doesn't give the greatest explanations nor the greatest examples (on the side note, half the time they never give examples or information to the questions later in the lesson) 

I'm just trying to get the process and theory of how to find such polynomial.

One more question is as follows:

One factor of 4x^3 + 15x^2 - 31x -30 is x - 2. Find the other factors.


Some help with these questions would be greatly appreciated seeing as I'm nearly done this course.

morgan · Accepted Answer

Problem #1
-----------
Given:
polynomial / (x+2) = (x^2 -4x + 1) with remainer 8

Therefore:
polynomial = (x^2 -4x + 1)*(x+2) + 8
polynomial = x^3 - 4x^2 +x + 2x^2 - 8x + 2 + 8
polynomial = x^3 - 2x^2 - 7x + 10


Problem #2
----------
One factor of 4x^3 + 15x^2 - 31x -30 is x - 2. Find the other
factors.

Step 1:
Divide x-2 into 4x^3 + 15x^2 + 31x - 30
The quotient is 4x^2 + 23x + 15

Then factor the quotient:
(x+5)(4x+3)

Thus, the three factors are:
(x-2)
(x+5)
(4x+3)

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Polynomial Division - Finding The Polynomial and Finding The Other Factors?

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