Find the dimension of each a The space of cubic polynomials

Find the dimension of each.

(a) The space of cubic polynomials p(x) such that p(7) = 0

(b) The space of cubic polynomials p(x) such that p(7) = 0 and p(5) = 0

a) Given that space of cubic polynomials p(x) such that p(7) = 0

Let p(x) = ax³+bx²+cx+d --------eq (1)

Given that p(7) = 0

a(7)³+ b(7)²+c.7 +d=0

243 a + 49b + 7c +d = 0

d = -243 a -49b - 7c

Substitute value of d in eq (1),

p(x) = ax³+bx²+cx -243 a -49b - 7c

p(x) = a (x³-243) +b (x²-49) +c (x-7)

Therefore,

dimension = { x-7 ,x²-49, x³-243 } = 3

(OR)

Given that p(7) = 0

Then,

p(x) = (x-7) (a+bx+cx²)

p1(x) = (x-5) , p2(x) = x (x-7) , p3(x) = x² (x-7) forms the basis.

Therefore,

dimension the space = 3

b) Given that

p(7) = 0 and p(5) = 0

Then,

p(x) = (x-5) (x-7) (a+bx)

p1(x) = (x-5) (x-7) , p2(x) = x (x-5) (x-7) forms the basis.

Therefore,

dimension the space = 2