What is the quotient (x³ + 8) ÷ (x + 2)?

Solution:

In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version of the familiar arithmetic technique called long division.

To find the quotient when (x³ + 8) is divided by (x + 2)

Dividend = (x³ + 8) , divisor = (x + 2)

Steps involved:

Divide the leading term of the dividend by the leading term of the divisor: x³/ x = x²

Multiply it by the divisor: x²(x + 2) = x³+ 2x²

Subtract the dividend from the obtained result: (x³+ 8) − (x³+ 2x²)

= −2x²+ 8

Divide the leading term of the obtained remainder by the leading term of the divisor: −2x²/ x = −2x

Multiply it by the divisor: -2x(x + 2)

= -2x²- 4x

Subtract the remainder from the obtained result: (-2x²+ 8) - (-2x²- 4x)

= 4x + 8

Divide the leading term of the obtained remainder by the leading term of the divisor: 4x / x = 4

Multiply it by the divisor: 4(x + 2) = 4x + 8

Since 4x + 8 is obtained from dividing the leading term of the obtained remainder by the divisor's leading term and previous step results.

Therefore, both the remainders cancel to give the ultimate remainder equal to 0.

Thus, one of the factors in x³ + 8 is x + 2 so its quotient will be 1.