Question

Find a basis {p(x),q(x)} for the vector space {f(x)∈P₂[x]∣f′(−8)=f(1)} where P₂[x] is the vector space of polynomials in x with degree at most 2 .

p(x)=
q(x)=

Answer 1

To find a basis for the given vector space, we need to find two polynomials p(x) and q(x) that satisfy the condition f'(−8) = f(1). The vector space consists of all polynomials of degree at most 2. Let's find p(x) and q(x) step-by-step.

Step-by-step explanation:

To find a basis for the given vector space, we need to find two polynomials p(x) and q(x) that satisfy the condition f'(−8) = f(1). The vector space consists of all polynomials of degree at most 2. Let's find p(x) and q(x) step-by-step:

1. Start with a general polynomial f(x) = ax^2 + bx + c.
2. Take the derivative of f(x): f'(x) = 2ax + b.
3. Substitute x = -8 into f'(x): f'(-8) = 2a(-8) + b = -16a + b.
4. Now, substitute x = 1 into f(x): f(1) = a(1)^2 + b(1) + c = a + b + c.
5. The condition f'(−8) = f(1) gives us the equation -16a + b = a + b + c.
6. Rearranging the equation, we get c = -17a.

So, any polynomial in the form f(x) = ax^2 + bx - 17a can be part of the vector space. To find a basis, we need to find two linearly independent polynomials that satisfy this condition. For example, we can choose p(x) = x^2 - 17 and q(x) = x. These two polynomials form a basis for the vector space.