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- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/06%3A_Products_and_Quotients_of_Groups/6.01%3A_Products_of_GroupsEvery finitely generated abelian group G is isomorphic to a direct product of cyclic groups of the form \[\mathbb{Z}_{p_1^{n_1}}\times \mathbb{Z}_{p_2^{n_2}}\times \cdots \times \mathbb{Z}_{p_r^{n...Every finitely generated abelian group G is isomorphic to a direct product of cyclic groups of the form Zpn11×Zpn22×⋯×Zpnrr×Zk, where each pi is a prime number (not necessarily distinct).
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Yet_Another_Introductory_Number_Theory_Textbook_-_Cryptology_Emphasis_(Poritz)/01%3A_Well-Ordering_and_Division/1.03%3A_Divisibility_and_the_Division_AlgorithmWe now discuss the concept of divisibility and its properties.
- https://math.libretexts.org/Courses/Coastline_College/Math_C097%3A_Support_for_Precalculus_Corequisite%3A_MATH_C170/1.04%3A_Polynomial_and_Rational_Functions/1.4.04%3A_Polynomial_DivisionFurthermore, the coefficients of the quotient polynomial match the coefficients of the first three terms in the last row, so we now take the plunge and write only the coefficients of the terms to get ...Furthermore, the coefficients of the quotient polynomial match the coefficients of the first three terms in the last row, so we now take the plunge and write only the coefficients of the terms to get To divide x3+4x2−5x−14 by x−2, we write 2 in the place of the divisor and the coefficients of x3+4x2−5x−14 in for the dividend.
- https://math.libretexts.org/Bookshelves/Algebra/Elementary_Algebra_(Arnold)/06%3A_Factoring/6.01%3A_The_Greatest_Common_FactorWe begin this section with definitions of factors and divisors. Because 24=2⋅12 , both 2 and 12 are factors of 24 . However, note that 2 is also a divisor of 24 , because when you divide 24 ...We begin this section with definitions of factors and divisors. Because 24=2⋅12 , both 2 and 12 are factors of 24 . However, note that 2 is also a divisor of 24 , because when you divide 24 by 2 you get 12 , with a remainder of zero. Similarly, 12 is also a divisor of 24 , because when you divide 24 by 12 you get 2 , with a remainder of zero.
- https://math.libretexts.org/Courses/Highline_College/MATHP_141%3A_Corequisite_Precalculus/02%3A_Algebra_Support/2.07%3A_The_Greatest_Common_Factor_and_Factor_by_Grouping\[2x(3x2+4x−7)=2x⋅3x2+2x⋅4x−2x⋅7=6x3+8x2−14x \nonumb...2x(3x2+4x−7)=2x⋅3x2+2x⋅4x−2x⋅7=6x3+8x2−14x 6x3+8x2−14x=2x⋅3x2+2x⋅4x−2x⋅7=2x(3x2+4x−7)
- https://math.libretexts.org/Courses/Coastline_College/Math_C097%3A_Support_for_Precalculus_Corequisite%3A_MATH_C170/1.02%3A_Algebra_Support/1.2.01%3A_Real_Numbers_Linear_Inequalities_and_Interval_NotationThe vertical line | inside the braces reads, “such that” and the symbol ∈ indicates set membership and reads, “is an element of.” The notation above in its entirety reads, “the set of all number...The vertical line | inside the braces reads, “such that” and the symbol ∈ indicates set membership and reads, “is an element of.” The notation above in its entirety reads, “the set of all numbers ab such that a and b are elements of the set of integers and b is not equal to zero.” Decimals that terminate or repeat are rational.
- https://math.libretexts.org/Courses/Highline_College/MATH_141%3A_Precalculus_I_(2nd_Edition)/03%3A_Polynomial_and_Rational_Functions/3.04%3A_Polynomial_DivisionFurthermore, the coefficients of the quotient polynomial match the coefficients of the first three terms in the last row, so we now take the plunge and write only the coefficients of the terms to get ...Furthermore, the coefficients of the quotient polynomial match the coefficients of the first three terms in the last row, so we now take the plunge and write only the coefficients of the terms to get To divide x3+4x2−5x−14 by x−2, we write 2 in the place of the divisor and the coefficients of x3+4x2−5x−14 in for the dividend.
- https://math.libretexts.org/Courses/Coastline_College/Math_C097%3A_Support_for_Precalculus_Corequisite%3A_MATH_C170/1.02%3A_Algebra_Support/1.2.07%3A_The_Greatest_Common_Factor_and_Factor_by_Grouping\[2x(3x2+4x−7)=2x⋅3x2+2x⋅4x−2x⋅7=6x3+8x2−14x \nonumb...2x(3x2+4x−7)=2x⋅3x2+2x⋅4x−2x⋅7=6x3+8x2−14x 6x3+8x2−14x=2x⋅3x2+2x⋅4x−2x⋅7=2x(3x2+4x−7)
- https://math.libretexts.org/Courses/Highline_College/MATHP_141%3A_Corequisite_Precalculus/02%3A_Algebra_Support/2.01%3A_Real_Numbers_Linear_Inequalities_and_Interval_NotationThe vertical line | inside the braces reads, “such that” and the symbol ∈ indicates set membership and reads, “is an element of.” The notation above in its entirety reads, “the set of all number...The vertical line | inside the braces reads, “such that” and the symbol ∈ indicates set membership and reads, “is an element of.” The notation above in its entirety reads, “the set of all numbers ab such that a and b are elements of the set of integers and b is not equal to zero.” Decimals that terminate or repeat are rational.
- https://math.libretexts.org/Courses/Highline_College/MATHP_141%3A_Corequisite_Precalculus/04%3A_Polynomial_and_Rational_Functions/4.04%3A_Polynomial_DivisionFurthermore, the coefficients of the quotient polynomial match the coefficients of the first three terms in the last row, so we now take the plunge and write only the coefficients of the terms to get ...Furthermore, the coefficients of the quotient polynomial match the coefficients of the first three terms in the last row, so we now take the plunge and write only the coefficients of the terms to get To divide x3+4x2−5x−14 by x−2, we write 2 in the place of the divisor and the coefficients of x3+4x2−5x−14 in for the dividend.
