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

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For more info, check out Mr. Buryn’s website

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)

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For more info, check out Mr. Buryn’s website

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

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For more info, check out Mr. Buryn’s website

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:

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For more info, check out Mr. Buryn’s website

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

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For more info, check out Mr. Buryn’s website

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

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For more info, check out Mr. Buryn’s website