Essential Mathematics: Rational Expressions

16 flashcards · Mathematics · by lakesare

[Rational expressions - distinct linear factors] How would you find the partial fraction expansion using the method of equating coefficients? You can always find the constants in the partial fraction expansion of a rational expression by using the method of equating coefficients – even rational expressions whose denominators are not a product of distinct linear factors. However, this method can be time-consuming, especially if there are three or more factors.
Write in A/factor_1 + B/factor_2 form Multiply by the denominator (x - 1)(x + 1) Equate coefficients
[Rational expressions] For rational expressions - what's denominators with distinct linear factors repeated linear factors quadratic factors?
distinct linear factors - 1st expression has distinct linear factors (x − 2) and (x + 3) in its denominator Partial fraction expansion can be found by the method of equating coefficients.repeated linear factors - 2nd expression has the repeated linear factor (x + 7). The factor is raised to a power greater than 1, Subsection 2.3.quadratic factors - 3rd expression has the quadratic factor (x2 + 1) in its denominator. Subsections 2.4.
[Rational expressions - repeated linear factors] What form of expansion will this rational expression have? Rational expressions can have denominators with distinct linear factors, repeated linear factors and quadratic factors.
Where A, B, C and D are constants.
[Rational expressions - distinct linear factors] What form of expansion will this rational expression have? Rational expressions can have denominators with distinct linear factors, repeated linear factors and quadratic factors.
[Rational functions] What's rational expression VS proper rational expression?
rational expression polynomial / polynomial proper rational expression polynomial of smaller degree / polynomial of larger degree Rational expression is an expression formed by dividing one polynomial expression by another. A rational function is a function whose rule is given by a rational expression. When you add or subtract two or more rational expressions you obtain another rational expression.
[Rational functions] Name 2 examples of improper rational expressions.
Improper rational expression are those in which the degree of the numerator is greater than or equal to the degree of the denominator.
[Rational functions] How to find a vertical asymptote?
Each number a for which the denominator of the rational expression is zero gives a vertical asymptote x = a.
[Rational functions] Name 3 types of asymptotes rational expressions can have. The graph of a function with a horizontal (y = 0), vertical (x = 0), and oblique asymptote (purple line, given by y = 2x).
horizontal Horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞. As the name indicates they are parallel to the x-axis.vertical Vertical asymptotes are vertical lines (perpendicular to the x-axis) near which the function grows without bound. slant Oblique asymptotes are diagonal lines such that the difference between the curve and the line approaches 0 as x tends to +∞ or −∞. The asymptotes most commonly encountered in the study of calculus…
[Rational functions] What's the dominant term of a polynomial?
The term in a polynomial with the highest power of x.
[Rational functions] In this equation - where is proper rational expression f(x) partial fraction expansion of f(x) partial fractions of f(x)
[Rational expressions] Why do we not discuss how to do the partial fraction expansion for rational expressions with (x8 + x3 + 2) in the denominator? Example of a rational expression.
Because any polynomial expression can be factorised into a product of linear and quadratic expressions.
[Rational functions] Name 4 possible forms of partial fraction expansion (of rational expression)? Example of a rational expression. A rational expression in x is an expression of the form a(x)/b(x) , where a(x) and b(x) are polynomial expressions.Deciding how to do the partial fraction expansion is about:1. the form of expansion2. and then about how to find the coefficients.
each linear factor (ax + b) gives the partial fraction A/(ax + b) each repeated linear factor (ax + b)n gives the partial fractions A1/(ax + b)1 +...+ An/(ax + b)n each quadratic factor (ax2 + bx + c) gives the partial fraction (Ax + b)/(ax2 + bx + c) If the quadratic factor has a zero or positive discriminant - then we should decompose it, and use linear factor or repeated linear factor forms.each repeated quadratic factor (ax2 + bx + c)n As you can see from this example, repeated quadratic fa…
[Rational functions] Name 3 possible ways of finding coefficients (of partial fractions)? ...of partial fractions of the partial fraction expansion of the rational function :-) Example of a rational expression. A rational expression in x is an expression of the form a(x)/b(x) , where a(x) and b(x) are polynomial expressions.Deciding how to do the partial fraction expansion is about:1. the form of expansion2. and then about how to find the coefficients.
method of equating coefficients method of substituting values cover-up method There is yet another method of finding the partial fraction expansion of a proper rational expression with distinct linear factors. It is essentially a streamlined version of the method of substituting values, which can be used when the values you are substituting are those that make the linear factors in the denominator of the rational expression equal 0.
[Rational functions] How to find the partial fraction expansion of the improper rational function?
Use polynomial division to actually divide numerator by denominator.
[Rational functions] In rational functions - how to find vertical VS horizontal & slant asymptotes? Example of a rational function
vertical asymptotes - set denominator to 0 and solve for x. As x approaches the value for the denominator is 0, f(x) becomes arbitrarily large.horizontal asymptotes - look at dominant terms. The degree (dominant term) of the numerator and degree of the denominator determine whether there are any horizontal or slant asymptotes. A rational function can have at most 1 horizontal or slant asymptote, and possibly several vertical asymptotes.
[Rational functions] In rational functions - how to find horizontal asymptotes and slant asymptotes? Example of a rational function A rational function can have at most 1 horizontal or slant asymptote, and possibly several vertical asymptotes.
If deg(a) - deg(b): < 0 - then y = 0 This is a horizontal asymptote.0 - then y = ratio of leading coefficients This is a horizontal asymptote.= 1 - then y = quotient of the polynomial division This is a slant asymptote.> 1 - none No linear asymptote, but a curvilinear asymptote exists. You can in fact use polynomial long division in this way to determine the equation of a horizontal asymptote and how the graph approaches it, but it’s usually simpler to use the methods described earlier.