choose a topic difference of cubes sum of cubes rational expressions and npv’s adding and...
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Difference of Cubes
Sum of Cubes
Rational Expressions and
NPV’s
Adding and Subtracting
Rational Expressions
Multiplying and Dividing Rational
Expressions
Difference of Cubes
This is a special law used to factor a difference of cubes binomial or a binomial made up of a perfect square being subtracted from a perfect square.
For example: 27-8,,
Practice
Identify the difference of squares binomials (click to check answers).
Difference of Cubes
The law works as follows:
Eg) The cube roots of the terms are: 2 and 4y
So, we put 2 in for all the x’s and 4y in for all the y’s
The answer would be:
Difference of Cubes
1. Find cube roots of both terms.
2. Swap all the x’s in with the cube root of the first term
3. Swap all the y’s with the cube root of the second term
4. Reduce all possible squares and multiplications
Practice
Factor these difference of cubes binomials
Practice
Answers
Difference of Cubes
Why does this work? Let’s reverse it to find out
(Cancel out)
Difference of Cubes Versus
Sum of cubes
Difference of cubes VersusSum of cubes
Since these two are so similar it is easy to mix them up, so here’s a trick to remember the difference
Look at the sign in the original binomial The first sign in the factors is the same The second sign is the opposite sign The third sign is always (+)a^3 ± b^3 = (a [same sign] b)(a^2 [opposite sign] ab [always positive] b^2)
Sum of Cubes
This is a law that is used to factor a sum of cubes binomial or a binomial in which both terms are perfect cubes and are being added together
Eg), ,
Practice
Identify the sum of cubes binomials
Sum of cubes
Step Example 1:
Example 2:
1: identify the cube roots of both terms
x and y 2 and 3y
2: substitute in one cube root for all the x’s and the other root for all the y’s
3: simplify (if possible)
Not possible to simplify
The law works as follows
Practice
Factor
Practice
Answers
Why it works
To see how the law works we can reverse the factoring:
Difference of CubesVersus
Sum of Cubes
Rational expressions
Rational expressions are basically fractions that have polynomials for numerators and denominators.
Eg)
Non-permissible Values
Non-permissible values (NPV’s) are any values for variables that will make a rational expression equal a non-real number
The most common type of NPV’s make the denominator equal to zero
Eg) for “x” has one NPV:
How to find NPV’s
Start by looking for any values for each variable that will the denominator equal to zero. Eg)
Now factor both the denominator and numerator fully
Look for any other NPV’s
Finding NPV’s
An easy way to look for numbers that make the denominator 0 is to remove the denominator and use algebra:
Eg)
Now use algebra:
or
Example
Find NPV’s: Factor Look for any new
NPV’s: Therefore the NPV’s are:
Adding and Subtracting Rational Expressions The first rule of adding and
subtracting rational expressions is to treat them just like regular fractions: First you use equivalent fractions to make both fractions have the same denominator and then you add or subtract the numerators without changing the denominators
How to guideStep
Example 1:
1: achieve common denominators(Factoring the denominators can help you do this if you are stuck)
=
2: add the numerators; the denominators stay the same
*The same method works for subtraction
Something to note…
Most questions will ask you to find the NPV’s. Remember to always look for NPV’s at EVERY step of the question: Before you do anything, after the denominators are the same, after you have added and subtracted the numerators, and even after you factor the final answer
Learn about NPV’s
Practice
Simplify through addition and subtraction.
Practice
Answers
Multiplying and Dividing Rational Expressions
The first rule for multiplying and dividing rational expressions is to treat them like fractions
Multiplication
First multiply the numerators using distributive law and simplify
Next multiply the denominators and simplify
Eg)
Division
First invert the second rational expression
Multiply the numerators and denominators
Simplify
Eg)
Something to note…
In many questions involving the multiplication or division of rational expressions, you will be asked to find NPV’s. Make sure that you look for NPV’s in ALL steps, before you multiply and divide and after. Even factor everything at the end to make sure that you found all of the NPV’s.
Learn about NPV’s
Practice
Simplify
Practice
Answers
