Final answer:
To divide the polynomials x^2 - 31x - 10 by x - 5, we perform polynomial long division. The result is x - 26 with a remainder of -130, giving us a final answer of x - 26 - 130/(x - 5).
Step-by-step explanation:
To divide polynomials, we use the method of long division. In this case, we are dividing the polynomial x^2 - 31x - 10 by the polynomial x - 5. Here are the steps:
Divide the first term of the numerator (x^2) by the first term of the denominator (x). This gives us x.
Multiply the entire denominator (x - 5) by the quotient we just found (x) to get x(x - 5) = x^2 - 5x.
Subtract this result from the numerator: (x^2 - 31x - 10) - (x^2 - 5x) = -26x -10.
Now, divide the first term of this new numerator (-26x) by the first term of the denominator (x). This gives us -26.
Multiply the entire denominator (x - 5) by the new quotient we found (-26) to get -26(x - 5) = -26x + 130.
Subtract this result from the numerator: (-26x - 10) - (-26x + 130) = -140.
So, the quotient is x - 26 and the remainder is -140. Therefore, the division of the polynomials is x - 26.