What will be the degree of the remainder if   `3y^(4)-6y^(2)-8y-5`   is divided by a quadratic polynomial?

Options

A. 3
B. 2
C. 1
D. 1 or 0

Answer

To find the degree of the remainder when the polynomial \(3y^4 - 6y^2 - 8y - 5\) is divided by a quadratic polynomial, we can follow these steps:

### Step 1: Identify the degree of the dividend and the divisor
The dividend is the polynomial \(P(y) = 3y^4 - 6y^2 - 8y - 5\). The degree of this polynomial is 4, as the highest power of \(y\) is 4.

The divisor is a quadratic polynomial, which can be represented as \(G(y) = ay^2 + by + c\). The degree of a quadratic polynomial is 2.

**Hint:** The degree of a polynomial is determined by the highest exponent of the variable in the polynomial.

### Step 2: Apply the Euclidean Division Algorithm
According to the Euclidean Division Algorithm, when a polynomial \(P(y)\) is divided by another polynomial \(G(y)\), we can express it as:
\[ P(y) = G(y) \cdot Q(y) + R(y) \]
where:
- \(Q(y)\) is the quotient,
- \(R(y)\) is the remainder.

### Step 3: Determine the degree of the remainder
The degree of the remainder \(R(y)\) must satisfy the following conditions:
- The degree of \(R(y)\) must be less than the degree of the divisor \(G(y)\).
- The degree of \(R(y)\) can be either 0 or 1 (since the degree of \(G(y)\) is 2).

Thus, the possible degrees of the remainder \(R(y)\) are:
- \( \text{Degree} \, R(y) < \text{Degree} \, G(y) \)
- This implies \( \text{Degree} \, R(y) < 2 \)

Therefore, the degree of the remainder \(R(y)\) can be:
- 0 (a constant) or
- 1 (a linear polynomial).

### Final Conclusion
The degree of the remainder when \(3y^4 - 6y^2 - 8y - 5\) is divided by a quadratic polynomial is either 0 or 1.

**Final Answer:** The degree of the remainder is either 0 or 1.