DIVIDE X^6 - 7X^3 - 8 BY THE PRODUCT OF X+1 AND (X^2 + 4) ???????
To divide the polynomial X^6 - 7X^3 - 8 by the product of X+1 and (X^2 + 4), we need to use polynomial long division. Follow these steps:
Step 1: Write the dividend and divisor in descending order of their powers, with missing terms represented by zero placeholders:
Dividend: X^6 - 7X^3 - 8
Divisor: (X + 1)(X^2 + 4)
Step 2: Perform the first division by dividing the highest power term of the dividend by the highest power term of the divisor. In this case, divide X^6 by X. The result is X^5.
Step 3: Multiply the quotient obtained in step 2 (X^5) by the entire divisor (X+1)(X^2 + 4). The result is X^5(X+1)(X^2 + 4).
Step 4: Subtract the result obtained in step 3 from the dividend. The subtraction process requires changing the signs of all the terms in the result obtained in step 3 and combining like terms. The result is a new polynomial.
Step 5: Repeat steps 2-4 with the new polynomial obtained in step 4 until you reach a remainder with a degree lower than the divisor.
By applying the long division process recursively, we get the quotient as X^5 - X^3 - 5X - 8, and the remainder as 7X - 7.
Therefore, the division of X^6 - 7X^3 - 8 by the product of X+1 and (X^2 + 4) is X^5 - X^3 - 5X - 8 with a remainder of 7X - 7.