Showing posts with label factor. Show all posts
Showing posts with label factor. Show all posts

Tuesday, November 6, 2012

Simplifying Rational Expressions

Given a rational expression, the quotient of two polynomials, we will factor the numerator and denominator if we can and then cancel factors that are exactly the same.
When evaluating rational expressions, plug in the appropriate values either before simplifying or after, the result will be the same.  Although, it is more efficient to simplify first then evaluate.
We can see that when evaluating, the result will be the same whether or not we simplify first.  It turns out that not all numbers can be used when we evaluate.
The point is that not all real numbers will be defined in the above rational expression.  In fact, there are two restrictions to the domain, -2 and 3/5.  These values, when plugged in, will result in zero in the denominator.  Another way to say this is that the domain consists of all real numbers except for −2 and 3/5.

Tip: To find the restrictions, set each factor in the denominator equal to zero and solve. The factors in the numerator do not contribute to the list of restrictions.

Simplify and state the restrictions to the domain.

Even if the factor cancels it still contributes to the list of restrictions.  Basically, it is important to remember the domain of the original expression when simplifying. Also, we must use caution when simplifying, please do not try to take obviously incorrect shortcuts like this:

Since subtraction is not commutative, we must be alert to opposite binomial factors.  For example, 5 − 3 = 2 and 3 − 5 = −2. In general,
Simplify and state the restrictions to the domain.
At this point, we evaluate using function notation.
Video Examples on YouTube:

Sunday, November 4, 2012

Solving Equations by Factoring

Previously we have learned how to solve linear equations, now we will outline a technique used to solve factorable quadratic equations that look like
In addition, we will revisit function notation and apply the techniques in this section to quadratic functions.
The above zero factor property is the key to solving quadratic equations by factoring. So far we have been solving linear equations, which usually had only one solution. We will see that quadratic equations have up to two solutions.


   Step 1: Obtain zero on one side and then factor.
   Step 2:  Set each factor equal to zero.
   Step 3: Solve each of the resulting equations.
This technique requires the zero factor property to work so make sure the quadratic is set equal to zero before factoring in step 1.
Tip: You can always see if you solved correctly by checking your answers. On an exam it is useful to know if got the correct solutions or not.

Instructional Video: The Zero-Product Property

When solving quadratic equations by factoring, the first step is to put the equation in standard form ax^2 + bx + c = 0, equal to zero.


Obtain standard form and then factor.
Set each factor equal to zero and solve.
Important: We must have zero on one side of our equation for this technique to work.

You can clear fractions from any equation by multiplying both sides by the LCD.

Multiply both sides by the LCD 6 here to clear the fractions.

Work the entire process in reverse to find equations given the solutions.

Find a quadratic equation with given solution set.
Remember that the notation y = f(x) reads, "y equals of x."
Evaluate the given function.
Given the graph of the quadratic function, find the x- and y- intercepts.
Video Examples on YouTube: