Divide as indicated.
(-2x5 - x3 + 4x2 + 58x - 20) ÷ (x2 - 5)
To divide the polynomial (-2x^5 - x^3 + 4x^2 + 58x - 20) by the polynomial (x^2 - 5), you can use polynomial long division.
Step 1: Arrange the polynomials in descending order of degree.
(-2x^5 - x^3 + 4x^2 + 58x - 20) ÷ (x^2 - 5)
Step 2: Divide the leading term of the dividend (-2x^5) by the leading term of the divisor (x^2). The result will be the first term of the quotient.
Quotient: -2x^3
Step 3: Multiply the divisor (x^2 - 5) by the first term of the quotient (-2x^3), and then subtract the result from the dividend.
-2x^3 * (x^2 - 5) = -2x^5 + 10x^3
-2x^5 - x^3 + 4x^2 + 58x - 20 - (-2x^5 + 10x^3)
Simplifying, we get:
-2x^5 - x^3 + 4x^2 + 58x - 20 + 2x^5 - 10x^3
Step 4: Combine like terms.
-2x^5 + 2x^5 - x^3 - 10x^3 + 4x^2 + 58x - 20
Simplifying, we get:
-11x^3 + 4x^2 + 58x - 20
Step 5: Repeat steps 2 to 4 until there are no more terms of higher degree in the dividend than the divisor.
Next, divide the leading term of the new dividend (-11x^3) by the leading term of the divisor (x^2).
Quotient: -11x
Multiply the divisor (x^2 - 5) by the new term of the quotient (-11x) and subtract the result from the dividend.
-11x * (x^2 - 5) = -11x^3 + 55x
-11x^3 + 4x^2 + 58x - 20 - (-11x^3 + 55x)
Simplifying, we get:
-11x^3 + 4x^2 + 58x - 20 + 11x^3 - 55x
Combine like terms.
-11x^3 + 11x^3 + 4x^2 + 58x - 55x - 20
Simplifying, we get:
4x^2 + 3x - 20
Step 6: At this point, the degree of the new dividend (4x^2) is lower than the degree of the divisor (x^2 - 5). Therefore, we stop here.
The final result of the division is:
Quotient: -2x^3 - 11x
Remainder: 4x^2 + 3x - 20