Question

The degree of the polynomial \((x^3 + x^2 + 2x + 1)(x^2 + 2x + 1)\) is

• A. 3
• B. 4
• C. 5
• D. 6

Hint:

To find the degree of a product of polynomials, you don't need to perform the full multiplication. Simply add the degrees of the highest power terms of each polynomial.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The degree of a polynomial is the highest exponent of its variable. When two polynomials are multiplied, the degree of the resulting polynomial is the sum of the degrees of the individual polynomials.

Step 2: Key Formula or Approach:
Let \(P(x)\) and \(Q(x)\) be two polynomials.
Degree of \((P(x) \times Q(x))\) = Degree of \(P(x)\) + Degree of \(Q(x)\).

Step 3: Detailed Explanation:
Let's identify the two polynomials:
First polynomial: \(P(x) = x^3 + x^2 + 2x + 1\).
The highest power of x in \(P(x)\) is 3. So, the degree of \(P(x)\) is 3.
Second polynomial: \(Q(x) = x^2 + 2x + 1\).
The highest power of x in \(Q(x)\) is 2. So, the degree of \(Q(x)\) is 2.
The degree of the product polynomial \((x^3 + x^2 + 2x + 1)(x^2 + 2x + 1)\) is the sum of their degrees.
\[ \text{Degree} = \text{Degree of } P(x) + \text{Degree of } Q(x) = 3 + 2 = 5 \]

Step 4: Final Answer:
The degree of the resulting polynomial is 5. This corresponds to option (C).
Alternatively, we can see that the term with the highest degree in the product will be obtained by multiplying the terms with the highest degrees from each polynomial: \(x^3 \times x^2 = x^{3+2} = x^5\). The degree is 5.