Find a factorable expression.
Which of the following quadratic polynomials can be expressed as a product of two linear factors with rational coefficients?
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1 solution
x 2 − 3 = ( x − 3 ) ( x + 3 )
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Factor them in the range of Rational numbers.
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You could mention that in the question
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@Jason Gomez – Formally, factoring means dividing the expressions with 2 or more than the factors, and until the rational number range.
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@. . – Technically, "factors" is not the same here as in number theory, which involves integers. See the examples here . While "factorization" involves factors in integer representations, that is not the rigorous definition. There exist some polynomials with recognizable patterns . Using one of the choices you listed, the simpler case is x 2 − 3 = ( x − 3 ) ( x + 3 ) . It is not factorizable in integer forms, but it is factorizable in real value forms. While the problem looks trivial, it isn't as all of the polynomials are factorizable in complex forms. That can easily be expressed by finding two roots (as that is the most possible number of roots of polynomials of degree 2 ) in the complex plane.
Also read this , which explores solutions of quadratic equations in a more complex perspective.
But did you got correct?
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Me? Yes I did but this is for others who come upon this question
Even 3 x 2 − 4 x − 1 , 5 x 2 − 7 x − 2 are factorable
All of these quadratics are factorable in the real numbers except for Choice D. You should consider restating the problem as which expression is irreducible (non-factorable) for real x.
Only 4 x 2 + 6 x + 2 is factorable because 4 x 2 + 6 x + 2 = ( 2 x + 1 ) ( 2 x + 2 ) = 2 ( x + 1 ) ( 2 x + 1 ) .