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  On the other hand, whenever a polynomial has been factored into only linear and irreducible quadratics, then it has been factored completely, since both linear factors and irreducible quadratics cannot be factored any further over the real numbers. 
  High school students' algebra experience should enable them to create and use tabular, symbolic, graphical, and verbal representations and to analyze and understand patterns, relations, and functions with more sophistication than in the middle grades. 
  In grades 9—12, students should develop an understanding of the algebraic properties that govern the manipulation of symbols in expressions, equations, and inequalities. 
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