Docsity
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Docsity AI
Docsity AI
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search for your university

Find the specific documents for your university's exams

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

Factoring Formulas: Perfect Square, Difference of Squares, and Difference & Sum of Cubes, Cheat Sheet of Mathematics

Michigan State University (MSU)Mathematics

Formulas and examples for factoring perfect squares, differences of squares, and differences and sums of cubes. It includes step-by-step solutions for various examples.

Typology: Cheat Sheet

2021/2022

Uploaded on 02/07/2022

ambau
ambau ๐Ÿ‡บ๐Ÿ‡ธ

4.5

(11)

250 documents

1 / 5

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
5.6 Special Factoring Formulas
A. Perfect Square Factoring
1. Perfect Square Factoring Formulas:
๎˜€๎˜ ๎˜‚ ๎˜ƒ๎˜€ ๎˜„ ๎˜‚ ๎˜„๎˜ ๎˜… ๎˜†๎˜€ ๎˜‚ ๎˜„ ๎˜‡๎˜
and
๎˜€๎˜ ๎˜ˆ ๎˜ƒ ๎˜€ ๎˜„ ๎˜‚ ๎˜„๎˜ ๎˜… ๎˜†๎˜€ ๎˜ˆ ๎˜„๎˜‡ ๎˜
2. To use: if the first and last terms of a trinomial are squares, try writing a perfect square;
then use the square formula to see if you are correct.
3. Examples:
Example 1: Factor
๎˜‰๎˜Š
๎˜ ๎˜‚ ๎˜‹ ๎˜ƒ
๎˜Š
๎˜‚ ๎˜Œ
.
Solution
Since
๎˜‰๎˜Š
๎˜๎˜… ๎˜†๎˜ƒ
๎˜Š
๎˜‡๎˜
and
๎˜Œ ๎˜… ๎˜ ๎˜
, we GUESS
๎˜†๎˜ƒ
๎˜Š
๎˜‚ ๎˜ ๎˜‡ ๎˜
Test: using the square formula,
๎˜†๎˜ƒ
๎˜Š
๎˜‚ ๎˜ ๎˜‡ ๎˜ ๎˜…
๎˜‰๎˜Š
๎˜ ๎˜‚ ๎˜‹ ๎˜ƒ
๎˜Š
๎˜‚ ๎˜Œ ๎˜Ž
Ans
๎˜†๎˜ƒ
๎˜Š
๎˜‚ ๎˜ ๎˜‡ ๎˜
Example 2: Factor
๎˜Œ
๎˜Š
๎˜ ๎˜ˆ ๎˜ƒ
๎˜‰๎˜Š ๎˜
๎˜‚ ๎˜‹ ๎˜
๎˜
๎˜
.
Solution
Since
๎˜Œ
๎˜Š
๎˜ ๎˜… ๎˜†๎˜
๎˜Š
๎˜‡๎˜
and
๎˜‹๎˜
๎˜
๎˜ ๎˜… ๎˜†
๎˜‰๎˜
๎˜‡๎˜
, we GUESS
๎˜†๎˜
๎˜Š
๎˜ˆ
๎˜‰ ๎˜
๎˜‡๎˜
Test: using the square formula,
๎˜†๎˜
๎˜Š
๎˜ˆ
๎˜‰ ๎˜
๎˜‡๎˜ ๎˜… ๎˜Œ
๎˜Š
๎˜ ๎˜ˆ ๎˜ƒ
๎˜‰๎˜Š ๎˜
๎˜‚ ๎˜‹ ๎˜
๎˜
๎˜ ๎˜Ž
Ans
๎˜†๎˜
๎˜Š
๎˜ˆ
๎˜‰ ๎˜
๎˜‡๎˜
1
pf3
pf4
pf5

Partial preview of the text

Download Factoring Formulas: Perfect Square, Difference of Squares, and Difference & Sum of Cubes and more Cheat Sheet Mathematics in PDF only on Docsity!

5.6 Special Factoring Formulas

A. Perfect Square Factoring

  1. Perfect Square Factoring Formulas : ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ and ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ 
  2. To use: if the first and last terms of a trinomial are squares, try writing a perfect square; then use the square formula to see if you are correct.
  3. Examples:

Example 1: Factor ^ ^ ^ ^.

Solution

Since ^ ^ ^ ^ ^ ^ and ^ ^ , we GUESS ^ ^ ^ ^ 

Test: using the square formula, ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ 

Ans ^ ^ ^ ^ 

Example 2: Factor ^ ^ ^  ^ ^  ^.

Solution

Since ^ ^ ^ ^ ^ and ^  ^ ^ ^  ^ ^ , we GUESS ^ ^  ^ 

Test: using the square formula, ^ ^  ^ ^ ^ ^ ^ ^  ^ ^  ^ 

Ans ^ ^  ^ 

Example 3: Factor ^ ^ ^ ^.

Solution

Since ^ ^ ^ ^ ^ ^ and ^ ^ , we GUESS ^ ^ ^ ^ 

Test: using the square formula, ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ X

This shortcut fails, so we must do AntiFOIL!

    TSP:   

Ans ^ ^ ^ ^ ^ 

B. Difference of Squares

  1. Formula : ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ 
  2. Examples:

Example 1: Factor ^ ^.

Solution

Write ^ ^ as ^ ^ 

By the formula, we get

Ans ^ ^ ^ ^ ^ 

C. Difference and Sum of Cubes

  1. Formulas

Difference of Cubes: ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ 

Sum of Cubes: ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ 

  1. Note : The quadratic in the factorization is prime (no need to try to factor it!)
  2. Easy way to remember these two formulas:

First factor : just โ€œremoveโ€ the cubes

Second factor : pretend to โ€œsquareโ€ the first factor EXCEPT rather than doing product times ^ , do product times 

  1. Examples:

Example 1: Factor ^ ^ ^ ^.

Solution

Write ^ ^ ^ ^ as ^ ^

By the formula, we get

Ans ^ ^ ^ ^ ^ ^ ^ 

Example 2: Factor ^ ^ ^ ^ ^.

Solution

Write ^ ^ ^ ^ ^ as ^ ^ ^ ^ ^ ^ ^ ^

By the formula, we get

Ans ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ 

D. Closing Comment

As always when factoring, you should first check to see if you can factor out a GCF before trying any other technique. In the last sections, we will put all of our techniques together.