Final answer:
To find the divisor polynomial g(x), we use the remainder theorem and express the given polynomial in terms of a quotient and remainder. After substituting the known quantities, we can simplify and rearrange to solve for g(x) as (x^4 + x^3 + x^2 + 11x + 10) divided by (x^2 - 2x + 5).
Step-by-step explanation:
When the polynomial x^4 + x^3 + x^2 + 10x - 2 is divided by a polynomial g(x), resulting in a quotient of x^2 - 2x + 5 and a remainder of -x - 12, we can find g(x) by using polynomial division. According to the remainder theorem, the polynomial can be expressed as:
p(x) = g(x) × q(x) + r(x), where:
- p(x) is the original polynomial
- g(x) is the divisor
- q(x) is the quotient
- r(x) is the remainder
Substituting the given values:
x^4 + x^3 + x^2 + 10x - 2 = g(x) × (x^2 - 2x + 5) + (-x - 12)
Now we can expand the right side of the equation and then compare coefficients to solve for g(x). After simplifying, we get:
g(x) × (x^2 - 2x + 5) = x^4 + x^3 + x^2 + 10x - 2 + x + 12
Rearranging the terms gives us:
g(x) × (x^2 - 2x + 5) = x^4 + x^3 + x^2 + 11x + 10
Finally, by dividing the left side of the equation by (x^2 - 2x + 5), we obtain that:
g(x) = (x^4 + x^3 + x^2 + 11x + 10) / (x^2 - 2x + 5)