Question

Given that the quotient of (x⁴+5x³−3x−15) divided by a polynomial is (x³−3), what is the polynomial?

a) x⁷+5x⁶−6x⁴−30x³+9x+45
b) x−5
c) x+5
d) x⁷+5x⁶+6x⁴+30x³+9x+45

Answer 1

To find the polynomial, use polynomial long division and set up the division as x³ - 3 | x⁴ + 5x³ - 3x - 15. Divide the first term of the dividend by the first term of the divisor, subtract the result, and repeat the process until the remainder is 0. The correct polynomial is x⁷ + 5x⁶ - 6x⁴ - 30x³ + 9x + 45, which corresponds to option a).

Step-by-step explanation:

To find the polynomial, we can use polynomial long division. Given that the quotient of (x4+5x3−3x−15) divided by a polynomial is (x3−3), we can set up the division as follows:

x³ - 3 | x4 + 5x³ - 3x - 15

First, divide the first term of the dividend by the first term of the divisor, which is x³ divided by x³. This gives us x. We then multiply the divisor by x to get x⁴ - 3x³. Subtract this from the dividend to get 8x³ - 3x - 15. Repeat the process until we have a remainder of 0. The final result is the polynomial x⁷ + 5x⁶ - 6x⁴ - 30x³ + 9x + 45. Therefore, the correct answer is option a).