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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

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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