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Difference of two squares.
The following column multiplication shows why the difference of two squares identity hold (C+A)(C-A) = C2 - A2 Column Multiplication C + A The identity
in equivalent form
shows how the difference of two squares may be factored. Example (I) - Applied to Numbers 36 - 16
Example (II) - Applied to Expressions x2 - 25
Example (III) - Applied to Expressions (2x-3)2 - 100
The Zero Product LawExercise - or food for thought.: Find the value of the missing digits ABC in the following puzzle 234 Assumption: If all the factors in a product are nonzero then the product is nonzero. The equivalent contrapositive form of this assumed implication rule is the following: Zero Product Law
The zero product law, its direct or indirect assumption, provides a reason for factoring quadratics and further polynomials p(x) for which the question where does p(x) = 0 is of interest. Two ways to solve equations:Example A.Suppose we want to solve
The latter was shown above in example II The solve an equation route would be to observe
The factorization route may be to observe that both factors in the product (x+5)(x-5) = x2 - 25 will be nonzero when x does not have the values + 5 or -5. The only way that one of the factors and hence the product can be zero is if x = 5 or x = -5. Example B.Suppose we want to solve 0 = (2x-3)2 - 100 = (2x+7)(2x-13) The latter was shown above in example III. The equation route to solve is say (2x-3)2 = 100 requires
Check that for x = 6.5 and x = -3.5, both satisfy 0 = (2x-3)2 - 100 or (2x-3)2 = 100 Now the factoring route observes (2x-3)2 - 100 = (2x+7)(2x-13) and that the product can only be zero if and only if 2x+ 7 = 0 or 2x - 13 = 0 or equivalently, if and only if, 2x = -7 or 2x = 13 or equivalently, if and only if, x = -7/2 = -3.5 or x = 13/2 = 6.5 This is the same result as before, except for the order of the solutions, an immaterial matter. In solving quadratic and other polynomial equations, you have a choice between the factorization route and the equation solving route. Choose the route that is most convenient if that is permitted by your instructor.
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