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 What must be subtracted from 4x^42x^36x^22x6 so that the result is exactly divisible by 2x^2x1?(x3 x2 x)(−x3 x2 x) = 2x2 2x Clearly 2x2 2x is not a polnomial of degree 3 and the second condition of a subspace is not satisfied c) The set of all polynomials p(x) in P 4 such that p(0) = 0 is a subspace of P 4 becuase it satisfies both conditions of a subspace To see this first note that all elements of the set described byStep by step solution Step 1 Trying to factor as a Difference of Squares 11 Factoring x 44 Theory A difference of two perfect squares, A 2 B 2 can be factored into (AB) • (AB) Proof (AB) • (AB) = A 2 AB BA B 2 = A 2 AB AB B 2 = A 2 B 2 Note AB = BA is the commutative property of multiplication Note AB AB equals zero and is therefore eliminated 3cx High Resolution Stock Photography And Images Alamy Sx4 pro guide spotting scope review

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Some moments are E X i = np i, VarX i = np i (1 − p i), and CovX i Xj = − np i p j The multinomial distribution generalizes the binomial Consider an experiment having a total of r possible outcomes, and let the corresponding probabilities be p 1, , p r, respectively Now perform n independent replications of the experiment and let Xi record the total number of times that the iR For example, X2 Xevaluates to 0 on Z 2,the field of integers modulo 2,since 11 = 0 mod 2 We will say more about evaluation maps in Section 25,when we study polynomial rings 6 If Ris a ring,thenRX,the set offormalpowerseries a 0 a 1X a 2X 2 ··· with coefficients in R,is also a ring under ordinary addition and multiplicationGives a unique quotient Uniqueness of r(x) follows from uniqueness of q(x) (3) Let X n be the n× nmatrix whose (i,j)entry is x i,j Show det(X n) is irreducible in Cx i,j 1≤i,j≤n SOLUTION Since the determinant is linear in rows, no monomial term of det(X n) contains a subfactor of the form x2 i,j or

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