Question 319069
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*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 24x^2y^3\ +\ 16xy^4\ +\ 48y^3]


The first term, *[tex \LARGE 24x^2y^3] which can be expressed as *[tex \LARGE 2\ \times\ 2\ \times\ 2\ \times\ 3\ \times\ x\ \times\ x\ \times\ y\ \times\ y\ \times\ y], has three factors of 2, one factor of 3, two factors of *[tex \Large x] and three factors of *[tex \Large y].


The second term, *[tex \LARGE 16xy^4] which can be expressed as *[tex \LARGE 2\ \times\ 2\ \times\ 2\ \times\ 2\ \times\ x\ \times\ y\ \times\ y\ \times\ y\ \times\ y], has four factors of 2, ZERO factors of 3, one factor of *[tex \Large x] and four factors of *[tex \Large y].


The third term, *[tex \LARGE 48y^3] which can be expressed as *[tex \LARGE 2\ \times\ 2\ \times\ 2\ \times\ 2\ \times\ 3\ \times\ y\ \times\ y\ \times\ y], has four factors of 2, one factors of 3, ZERO factors of *[tex \Large x] and three factors of *[tex \Large y].


The fewest times that the factor 2 appears in any of the terms is 3 times.


The fewest times that the factor 3 appears in any of the terms is ZERO times.


The fewest times that the factor *[tex \Large x] appears in any of the terms is ZERO times.


The fewest times that the factor *[tex \Large y] appears in any of the terms is 3 times.


Hence, the largest factor that is in common with all three terms, i.e. the greatest common factor or GCF, is *[tex \Large 2^3y^3\ =\ 8y^3]



*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 24x^2y^3\ \div\ 8y^3\ =\ 3x^2]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 16xy^4\ \div\ 8y^3\ =\ 2xy]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 48y^3\ \div\ 8y^3\ =\ 6]


Hence:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 24x^2y^3\ +\ 16xy^4\ +\ 48y^3\ =\ 8y^3\left(3x^2\ +\ 2xy\ +\ 6\right)]


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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