Rational Expressions
Simplifying Rational Expressions
Property 4.1
Property 4.2 Fundamental Principle of Fractions
If \(b\) and \(k\) are nonzero integers and \(a\) is any integer, then
$$ \frac{a \cdot k}{b \cdot k} = \frac{a}{b} $$More on Rational Expressions and Complex Fractions
Simplifying Complex Fractions
Definition
Complex fractions are fractional forms that contain rational numbers or rational expressions in the numerators and/or denominators.
Dividing Polynomials
Given a polynomial \(p(x)\) and a divisor of the form \(x - k\) the division process can be simplified by a procedure called synthetic division. This procedure is a shortcut for this type of polynomial division.
First, let’s consider an example and use the usual division process. Then, in step-by-step fashion, we can observe some shortcuts that will lead us into the synthetic-division procedure. Consider the division problem
$$ (3x^3 - 2x^2 + 6x - 5) / (x + 4) $$- Write the coefficients of the dividend as follows
- In the divisor, \(x + 4\), use (-4) instead of (4) so that later we can add rather than subtract.
- Bring down the first coefficient of the dividend \((3)\).
- Multiply \((3)(-4)\), which yields \(-12\); this result is to be added to the second coefficient of the dividend \(-2\).
- Multiply \((-14)(-4)\), which yields \(56\); this result is to be added to the third coefficient of the dividend \((6)\).
- Multiply \((62)(-4)\), which yields \(-248\); this result is added to the last term of the dividend \((-5)\).
The last row indicates a quotient of \(3x^2 - 14x + 62\) and a remainder of \(-253\).
Fractional Equations
Defitinion
A statement of equality between two ratios is called a proportion.