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are solved using cross-multiplication, since the missing b term is implicitly equal to 1: =. Any equation containing fractions or rational expressions can be simplified by multiplying both sides by the least common denominator. This step is called clearing fractions.
Because of this, often, the only simple effective way to deal with multiplication by expressions involving variables is to substitute each of the solutions obtained into the original equation and confirm that this yields a valid equation. After discarding solutions that yield an invalid equation, we will have the correct set of solutions.
The FOIL method is a special case of a more general method for multiplying algebraic expressions using the distributive law. The word FOIL was originally intended solely as a mnemonic for high-school students learning algebra. The term appears in William Betz's 1929 text Algebra for Today, where he states: [2]
A rational algebraic expression (or rational expression) is an algebraic expression that can be written as a quotient of polynomials, such as x 2 + 4x + 4. An irrational algebraic expression is one that is not rational, such as √ x + 4.
In mathematics, the method of clearing denominators, also called clearing fractions, is a technique for simplifying an equation equating two expressions that each are a sum of rational expressions – which includes simple fractions.
There are a number of ways the constants can be found. The most straightforward method is to multiply through by the common denominator q(x). We then obtain an equation of polynomials whose left-hand side is simply p(x) and whose right-hand side has coefficients which are linear expressions of the constants A ir, B ir, and C ir. Since two ...
Multiplication symbols are usually omitted, and implied, when there is no operator between two variables or terms, or when a coefficient is used. For example, 3 × x 2 is written as 3x 2, and 2 × x × y is written as 2xy. [5] Sometimes, multiplication symbols are replaced with either a dot or center-dot, so that x × y is written as either x ...
Systematic generalizations of this basic definition define the multiplication of integers (including negative numbers), rational numbers (fractions), and real numbers. Multiplication can also be visualized as counting objects arranged in a rectangle (for whole numbers) or as finding the area of a rectangle whose sides have some given lengths.
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