Question

When x^4 +x^3 + x^2 +10x -2 is divided by a polynomial g(x), the quotient and the remainder are x^2 -2X +5 and -x-12 respectively. Find g(x).

Answer 1

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)