Trigonometry!

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Unit 3: Rational Equations

3.1 Reciprocal of a Linear Function

• The reciprocal of a linear function has the form:

                     f(x) = 1 / kx – c

• The restriction on a domain of a reciprocal linear function can be determined by finding the value of x that makes the denominator equal to zero, that is x = c / k.
• The Vertical Asymptote of a reciprocal linear function has an equation of the form x = k / c. 
• The horizontal asymptote of a reciprocal linear function has equation y = 0.
• If k > 0, the left branch of a reciprocal linear function has a negative, decreasing slope, and the right branch has a negative, increasing slope.
• If k < 0, the left branch of a reciprocal linear function has a positive, increasing slow, and the right branch has a positive, decreasing slope.

 3.2 Reciprocal of a Quadratic Function

• Rational quadratic functions can be analyzed using key features:asymptotes, intercepts, slope (positive or negative, increasing or decreasing), domain, range, and positive and negative intervals.
• Reciprocal of quadratic functions with two zeros have three parts, with the middle one reaching a maximum or minimum points. This point is equidistant from the two vertical asymptotes.
• The behavior near asymptotes is similar to that of reciprocals of linear functions.
• All of the behaviors listed above can be predicted by analyzing the roots of the quadratic relation to the denominator.

 3.3 Rational Functions of the Form  f(x) = (ax + b) / (cx + d) 

• A rational function of the form f(x) = (ax + b) / (cx + d) has the following key features:
• The vertical asymptote can be found by setting the denominator equal to zero and solving for x, provided the numerator does not have the same zero.
• The horizontal asymptote can be found by dividing each term in both the numerator and the denominator by x and investigating the behavior of the function as x -> positive or negative infinity.
• The coefficient b acts to stretch the curve but has no effect on the asymptotes, domain, or range.
• The coefficient d shifts the vertical asymptote.
• The two branches of the graph of the function are equidistant from the point of intersection of the vertical and horizontal asymptotes.
• Analysis of End Behavior
• For vertical asymptote
• Substitute a number very close to the VA from the right, and a number from the left
• Analyze the result of that number and express the end behavior
• Whether As x -> VA +/- , y -> +/- infinity
• For horizontal asymptote
• Substitute a very large negative and positive number for x and analyze the behavior of y.
• Express the end behavior with the results from that substitution
• As x -> +/- Infinity, y -> HA from above/below

 3.4 Solve Rational Equations and Inequalities

• To solve rational equations algebraically, start by factoring the expressions in the numerator and denominator to find asymptotes and restrictions.
• Next, multiply both sides by the factored denominators, and simplify to obtain a polynomial equation. Then solve.

3.5 Making Connections With Rational Functions and Equations 

• For Rational inequalities
• Set the right side of the equation zero.
• Factor the expression to find restrictions
• Based on the assumption that x = a / b is true if and only if a * b = x.
• On the left side of the equation, take the denominator and multiply it by the numerator.
• Since the equation is already factored, the roots are clearly shown. Graph or use Interval Table method to find the intervals x which satisfy the equation

Advanced Function Chapter 2: Polynomial Equations and Inequalities

Long Term Division

Find

The problem is written like this:

The quotient and remainder can then be determined as follows:

1. Divide the first term of the numerator by the highest term of the denominator (meaning the one with the highest power of x, which in this case is x). Place the result above the bar (x³ ÷ x = x²).

   

2. Multiply the denominator by the result just obtained (the first term of the eventual quotient). Write the result under the first two terms of the numerator (x² · (x − 3) = x³ − 3x²).

   

3. Subtract the product just obtained from the appropriate terms of the original numerator (being careful that subtracting something having a minus sign is equivalent to adding something having a plus sign), and write the result underneath ((x³ − 12x²) − (x³ − 3x²) = −12x² + 3x² = −9x²) Then, "bring down" the next term from the numerator.

   

4. Repeat the previous three steps, except this time use the two terms that have just been written as the numerator.

   

5. Repeat step 4. This time, there is nothing to "pull down".

   

The polynomial above the bar is the quotient, and the number left over (−123) is the remainder.

The log division algorithm for arithmetic can be viewed as a special case of the above algorithm, in which the variable x is replaced by the specific number 10.

 Remainder Theorem

When you divide a polynomial P(x) by x-b you get:

p(x) = (x-b)·q(x) + r(x)

But r(x) is simply the constant r (remember? when you divide by (x-b) the remainder is a constant) .... so we get this:

f(x) = (x-b)·q(x) + r

Now see what happens when you have x equal to b:

p(b) = (b-b)·q(b) + r

p(b) = (0)·q(b) + r

p(b) = r

So we get this:

The Remainder Theorem:

When you divide a polynomial p(b) by x-b the remainder r will be p(b)

So if you want to know the remainder after dividing by x-b you don't need to do any division:

Just calculate p(b).

Let us see that in practice:

Example: 2x²-5x-1 divided by x-3

(Continuing our example from above)

We don't need to divide by (x-3) ... just calculate f(3):

2(3)²-5(3)-1 = 2x9-5x3-1 = 18-15-1 = 2

And that is the remainder we got from our calculations above.

We didn't need to do Long Division at all!

Example: Dividing by x-4

(Continuing our examplee)

What would the remainder be if we divided by "x-4" ?

"c" is 4, so let us check f(4):

2(4)²-5(4)-1 = 2x16-5x4-1 = 32-20-1 = 11

Once again ... We didn't need to do Long Division to find it.

 Factor Theorem

Now ...

What if you calculated p(b) and it was 0?

... that means the remainder is 0, and ...

... (x-b) must be a factor of the polynomial!

Example: x²-3x-4

f(4) = (4)²-3(4)-4 = 16-12-4 = 0

so (x-4) must be a factor of x²-3x-4

And so we have:

The Factor Theorem:

When p(b)=0 then x-b is a factor of the polynomial

And the other way around, too:

When x-b is a factor of the polynomial then p(b)=0

Polynomial Equations

• Real roots of a polynomial equation P(x) = 0 correspond to the x-intercepts of the graph of the polynomial function P(x).
• X-intercepts of the graph of a polynomial function correspond to the real roots of the related polynomial equation.
• If a polynomial equation is factorable, factoring the polynomial, setting its factors equal to zero, and solving each factor will determine the roots.

Families of a Polynomial Function

• A family of functions is a set of functions with the same characteristics.
• Polynomial functions with graphs that have the same x-intercepts belong to the same family.
• A family of polynomial functions with zeros a₁,a₂,a₃,…a_n can be represented by an equation of the form:

y = k(x – a₁)(x – a₂)(x – a₃) . . . (x – a_n), where k is a real number not equal to zero

• An equation for a particular member of a family of polynomial functions can be determined if a point on the graph is known.

Solving Inequalities or Inequations

• A polynomial inequality results when the equal sign in a polynomial equation is replaced with an inequality symbol.
• The real zeros of a polynomial function, or x-intercepts of the corresponding graph, divide the x-axis into intervals that can be used to solve a polynomial inequality.
• Polynomial inequalities may be solved graphically by determining the x-intercepts and then using the graph to determine the intervals that satisfy the inequality.
• Factoring inequalities can be solved algebraically by:
• Considering all cases
• Using intervals and testing values in each interval
• Table and number lines can help organize intervals and to provide a visual clue to solutions.