によって David Kedrowski 14年前.
328
MAT.116 1.3-1.5
Rational expressions are essentially fractions where both the numerator and the denominator are polynomials. The key operations on these expressions—multiplication, division, addition, and subtraction—
によって David Kedrowski 14年前.
328
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If a is a real number and n is a natural number, then
a^n = a*a*a* . . . *a.
The natural number n is called the exponent (or power) and the real number a is called the base.
Takes a mix of the stated properties.
If a and b are nonzero real numbers and m and n are integers, then
If a is any nonzero real number and n is a positive integer, then
a^(-n) = 1 / a^n
For any nonzero real number a, a^0=1.
0^0 is indeterminate.
A compound rational expression is a rational expression that has a rational expression in its numerator and/or denominator.
Simplifying a compound rational expression typically requires a combination of the four basic operations plus factoring.
It often helps to rewrite compound rational expressions in division form with the division symbol.
This is the same as for fractions: find a common denominator for the rational expressions, subtract the second numerators from the first (keeping the common denominator), and then simplify the resulting quotient.
It is typically easiest to leave the common denominator in factored form to help see which factors can be cancelled after the subtraction has been performed.
This is the same as for fractions: find a common denominator for the rational expressions, add the numerators (keeping the common denominator), and then simplify the resulting quotient.
It is typically easiest to leave the common denominator in factored form to help see which factors can be cancelled after the addition has been performed.
This is the same as for fractions: multiply the first rational expression by the reciprocal of the second and then simplify the resulting quotient.
It is typically easiest to factor the rational expressions before multiplying, and then leave everything factored to help see which factors can be cancelled.
This is the same as for fractions: multiply straight across and then simplify the resulting quotient.
It is typically easiest to factor the rational expressions before multiplying, and then leave everything factored to help see which factors can be cancelled.
A rational expression is simplified, or reduced to lowest terms, if its numerator and denominator have no common factors other than 1 (and -1).
We can only cancel common factors, we cannot cancel common (or even similar) terms.
Other Methods
There is a more mechanical method that can be used in place of the trial-and-error method for quadratic polynomials with leading coefficient not 1 and linear coefficient not 0.
There are also graphical techniques for finding factorizations. These involve treating the polynomial as a function, graphing it, and looking for the zeros of the function (also known as the x-intercepts or roots).
Factoring by Regrouping
Sometimes when a polynomial has four terms it is possible to factor the terms in pairs and find a common factor to each pair. Then the distributive law can be used to finish the factoring process.
Trial-and-Error Factorization
For quadratic polynomials of the form x^2 + bx + c, where b is not 0, we can factor by finding two factors for c which sum to b.
Leading Coefficient Not 1
For quadratic polynomials of the form ax^2 + bx + c, where a and b are not 0, we can factor using a trial-and-error method.
First, choose an integer factorization for the leading coefficient a. For example, let pq = a be such a factorization.
Then, if our factorization of a is correct, we know that the factored form must look like (px + r)(qx + s), where rs = c. We try different values for r and s to see if we can find a combination, along with p and q, that end up giving b for the coefficient of the middle term.
If no such combination of r and s can be found, we choose a different factorization for the leading coefficient a and repeat the process.
If, after trying all possible combinations of p and q values with all possible combinations of r and s values a factorization is never found, it can be stated that the polynomial is prime (over the integers).
Leading Coefficient 1
Some Important Formulas
Difference of Two Squares
Pefect Square Trinomial
In the text, see Table 7.
Don't worry about the cube formulas.
Common Factors
The first step whenever factoring a polynomial is to factor out the greatest common factor from all of the terms.
Factoring
Expressing a polynomial as a product of two (or more) polynomials, each of lesser degree than the original polynomial.
A polynomial is completely factored (over the set of integers) if it is expressed as a product of prime polynomials with integral coefficients.
NOTE: The text's author has chosen to limit factoring to the set of integers. It is also possible, though considerably more difficult, to factor over the set of rational numbers, or even over the set of real numbers.
A rational expression is a quotient of polynomials.
That is, it is a ratio (or fraction) of polynomials.
The denominator can never be 0.
