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2. Let P_3 be the vector space of all polynomials of degree 2 or less. We can write this as P_3 = {ax^3 + bx^2 + cx + d | a, b, c, d ? R} . Let W be the subset W = {ax^3 + ax^2 + ax + a | a ? R} . a) Write down a polynomial p(x) that is in W. b) Write down a polynomial q(x) that is in P_3 but not in W. c) Is W a subspace of P_3? If yes, show that it is closed under addition and scalar multiplication. If not, show that it fails one of these. 3. Repeat all three parts of Exercise #2, but use the subset X of P_3 given by X = {ax^3 + ax^2 + 10ax + 10a | a ? R} . 4. Repeat all three parts of Exercise #2, but use the subset U of P_3 given by U = {ax^3 + ax^2 + (10 + a)x + a | a ? R} .

          2. Let P_3 be the vector space of all polynomials of degree 2 or less. We can write this as
P_3 = {ax^3 + bx^2 + cx + d | a, b, c, d ? R} .
Let W be the subset
W = {ax^3 + ax^2 + ax + a | a ? R} .
a) Write down a polynomial p(x) that is in W.
b) Write down a polynomial q(x) that is in P_3 but not in W.
c) Is W a subspace of P_3? If yes, show that it is closed under addition and scalar multiplication. If not, show that it fails one of these.

3. Repeat all three parts of Exercise #2, but use the subset X of P_3 given by
X = {ax^3 + ax^2 + 10ax + 10a | a ? R} .

4. Repeat all three parts of Exercise #2, but use the subset U of P_3 given by
U = {ax^3 + ax^2 + (10 + a)x + a | a ? R} .

2. Let P3 be the vector space of all polynomials of degree 2 or less. We can write this as
P3 = ax^3 + bx^2 + cx + d | a, b, c, d ? R .
Let W be the subset
W = ax^3 + ax^2 + ax + a | a ? R .
a) Write down a polynomial p(x) that is in W.
b) Write down a polynomial q(x) that is in P3 but not in W.
c) Is W a subspace of P3? If yes, show that it is closed under addition and scalar multiplication. If not, show that it fails one of these.

3. Repeat all three parts of Exercise #2, but use the subset X of P3 given by
X = ax^3 + ax^2 + 10ax + 10a | a ? R .

4. Repeat all three parts of Exercise #2, but use the subset U of P3 given by
U = ax^3 + ax^2 + (10 + a)x + a | a ? R .

Added by Jessica K.

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