What are the ways of reducing rational expressions to lowest terms?


Quick Answer

Rational expressions are reduced to their simplest forms by factoring. Factoring allows the removal of the greatest common factors, which makes the expression have lower terms and not look as intimidating. Several factoring methods exist. Factoring by grouping and by removing the greatest common factors are the two most prominent.

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Full Answer

Factoring by grouping involves rearranging the expressions by degree, grouping them into smaller parenthetical expressions and simplifying them. After removing common terms, the common terms are placed in their own group. For example, the expression x^4 + 3x^3 - x^2 + 5x simplifies to x^3(x + 3) - x(x + 5). In turn, this goes to (x^3 - x)(x + 3)(x + 5).

Factoring with the distributive property works similarly. For example, the expression (6x^2 -12x + 9)/(3x - 6) reduces to 3(x - 1)(x - 3)/3(x - 2). The 3 cancels out, leaving (x - 1)(x - 3)/(x - 2). This expression can be placed on a graph, leaving all real solutions except for 2, because 2 would make the denominator zero.

Radicals, such as square or cube roots, cannot occupy the denominator. If one occurs, the whole expression has to be squared or cubed to cancel out the radical and leave the expression in its simplest form.

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