Let P3 be the set of polynomials of degree 3 or less with...

Let P3 be the set of polynomials of degree 3 or less with coefficients from ?. Let W = {p(x) | p(x) ? P3 and p(5) = 0} That is the set of all polynomials from P3 that pass through the point (5, 0). Show that W is a subspace of P3 (or that it's not). Recall this requires showing three things: 1. W contains the zero vector 2. W is closed under addition: for all ?u, ?v ? W, ?u + ?v ? W 3. W is closed under scalar multiplication: for all ?u ? W and k ? ?, k?u ? W

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00:01 Hi everyone in this question we have given a set w which is nothing but collection of all the polynomials such that that polynomial is a polynomial of degree 3 which is denoted by p3 and also it is such that p of 5 is equal to 0 that 5 is the root of this polynomial okay then we have to check in this question that whether this is a subspace of v or not what is v is the vector space of all the polynomials okay we have to check whether it is a subspace or not for subspace the first thing we have to check is that whether it contains zero or not okay so let zero be equal to that of b of x is equal to zero plus zero plus 0x square and so on till it's basically you can say 0 polynomial which will be looking like 0 plus 0x plus 0x square plus 0 x x raised to 3 because we are collecting all the polynomials of degree 3 okay so if you put this polynomial if you put p of 5 if you consider this as your zero polynomial if you consider p of 5 then it definitely give you 0 okay even after putting x is equal to 5 you will definitely get p of 5 is equal to 0 this this implies that 0 belongs to this w okay i can write that w contains 0 contains 0 vector 0 vector now i just wanted to show now whether it is closed and tradition or not.

02:17 Okay, for that what i will do let p of x and q of x belongs to w.

02:29 Okay? p of x belongs to w which implies that p of 5 is equal to 0.

02:38 And q of x belongs to w.

02:42 It implies that q of 5 is equal to 0.

02:47 So consider p plus q of 5, which will be nothing but p of 5 plus q of 5, right? but this we know that p of 5 is 0, q of 5 is 0.

03:05 So it implies that p plus q of 5 of 5 is equal to 0...

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