Questions: Determine whether the given set S of vectors is closed under addition and closed under scalar multiplication. The set S of all polynomials of degree exactly 2.

Determine whether the given set S of vectors is closed under addition and closed under scalar multiplication. The set S of all polynomials of degree exactly 2.

Transcript text: Determine whether the given set s of vectors is closed under addition and closed under scalar multiplication 13. The set $S$ of all polynomials of degree exactly 2 .

Solution

Solution Steps

To determine if the set $ S $ of all polynomials of degree exactly 2 is closed under addition and scalar multiplication, we need to check two properties:

Closure under addition: For any two polynomials $ p(x) $ and $ q(x) $ in $ S $, their sum $ p(x) + q(x) $ should also be in $ S $.
Closure under scalar multiplication: For any polynomial $ p(x) $ in $ S $ and any scalar $ c $, the product $ c \cdot p(x) $ should also be in $ S $.

For polynomials of degree exactly 2, the general form is $ ax^2 + bx + c $ where $ a \neq 0 $.

Solution Approach

Addition: Check if the sum of two polynomials of degree exactly 2 results in a polynomial of degree exactly 2.
Scalar Multiplication: Check if multiplying a polynomial of degree exactly 2 by a scalar results in a polynomial of degree exactly 2.

Step 1: Define the Set of Polynomials

The set $ S $ consists of all polynomials of degree exactly 2. A polynomial of degree exactly 2 can be expressed in the form $ ax^2 + bx + c $ where $ a \neq 0 $.

Step 2: Check Closure Under Addition

To determine if $ S $ is closed under addition, consider two polynomials $ p_1(x) = a_1x^2 + b_1x + c_1 $ and $ p_2(x) = a_2x^2 + b_2x + c_2 $. Their sum is: \[ p_1(x) + p_2(x) = (a_1 + a_2)x^2 + (b_1 + b_2)x + (c_1 + c_2) \] For the sum to be a polynomial of degree exactly 2, the coefficient of $ x^2 $ must be non-zero, i.e., $ a_1 + a_2 \neq 0 $. Given $ a_1 = 1 $ and $ a_2 = 2 $, we have $ a_1 + a_2 = 3 \neq 0 $. Thus, $ S $ is closed under addition.

Step 3: Check Closure Under Scalar Multiplication

To determine if $ S $ is closed under scalar multiplication, consider a polynomial $ p(x) = ax^2 + bx + c $ and a scalar $ k $. The product is: \[ k \cdot p(x) = (k \cdot a)x^2 + (k \cdot b)x + (k \cdot c) \] For the product to be a polynomial of degree exactly 2, the coefficient of $ x^2 $ must be non-zero, i.e., $ k \cdot a \neq 0 $. Given $ a = 1 $ and $ k = 5 $, we have $ k \cdot a = 5 \neq 0 $. Thus, $ S $ is closed under scalar multiplication.