Questions: Divide using the division algorithm. Write your answer in the form Q+R/D where the degree of R is less than the degree of D. (y^3-8)/(y-2)

Transcript text: Divide using the division algorithm. Write your answer in the form $Q+\frac{R}{D}$ where the degree of $R$ is less than the degree of $D$. \[ \frac{y^{3}-8}{y-2} \]

Solution

Solution Steps

To divide the polynomial $ y^3 - 8 $ by $ y - 2 $ using the division algorithm, we perform polynomial long division. The result will be expressed in the form $ Q + \frac{R}{D} $, where $ Q $ is the quotient and $ R $ is the remainder. The degree of $ R $ should be less than the degree of $ D $.

Step 1: Polynomial Division

We need to divide the polynomial $ y^3 - 8 $ by $ y - 2 $. Using polynomial long division, we find the quotient and remainder.

Step 2: Finding the Quotient

The quotient obtained from the division is $ y^2 + 2y + 4 $.

Step 3: Finding the Remainder

Since the degree of the remainder must be less than the degree of the divisor $ y - 2 $, and in this case, the remainder is $ 0 $, we can express the result of the division as: \[ \frac{y^3 - 8}{y - 2} = y^2 + 2y + 4 + \frac{0}{y - 2} \]