Wednesday, January 3, 2018

Middle School Math Solutions – Expand Calculator, Binomial Expansion

We’ve learned how to expand perfect squares and perfect cubes. Now, we are going to learn how to expand binomials raised to any positive integers.

Imagine having to expand a binomial raised to a power of 7. Sounds like a lot of work, right? Good news, we have a formula!

Here is the Binomial Theorem:


\binom{n}{i} is a combination, which we read as “n choose i”.
Here is the formula for n choose i:


Let’s see some examples using this formula.

First example (click here):

                                                          Expand (x+2)^4

1. Apply the formula

                                                         a=x, b=2, n=4


2. Expand the summation and simplify

           \sum_{i=0}^4\binom{4}{i}x^(4-i)2^i =\binom{4}{0} x^4∙2^0+\binom{4}{1} x^3∙2^1+\binom{4}{2} x^2∙2^2+\binom{4}{3} x^1 ∙2^3+\binom{4}{4} x^0∙2^4

                                       =\frac{4!}{0!(4-0)!} x^4∙2^0+\frac{4!}{1!(4-1)!} x^3∙2^1 
                                          +\frac{4!}{2!(4-2)!} x^2∙2^2+\frac{4!}{3!(4-3)!} x^1 ∙2^3+\frac{4!}{4!(4-4)!} x^0∙2^4
                                 =\frac{24}{(1)24} x^4∙2^0+\frac{24}{1(6)} x^3∙2^1+\frac{24}{2(2)}x^2∙2^2+\frac{24}{6(1)} x^1 ∙2^3+\frac{24}{24(1)} x^0∙2^4

You can see that there is a lot to calculate, but that this formula makes expanding easier and faster. In this example, I went into detail on how to simplify this expansion, specifically the combination. Now, that you’ve seen and understand how to calculate combinations, the next examples won’t be in such detail (you can also check out a more detailed step by step solution by clicking the hyperlinks).

Next example (click here):

                                                             Expand (x-y)^5

1. Apply the formula

                                                              a=x, b=-y, n=5

                                           (x-y)^5=\sum_{i=0}^5\binom{5}{i} x^(5-i) (-y)^i

2. Expand the summation and simplify

               \sum_{i=0}^5\binom{5}{i} x^(5-i) (-y)^i
                                  =\binom{5}{0} x^5∙(-y)^0+\binom{5}{1} x^4∙(-y)^1+\binom{5}{2} x^3∙(-y)^2+\binom{5}{3} x^2 ∙(-y)^3+\binom{5}{4} x^1∙(-y)^4+\binom{5}{5} x^0 (-y)^5
               =x^5-5x^4 y+10x^3 y^2-10x^2 y^3+5xy^4-y^5

Last example (click here):

                                                          Expand (3+x^2 )^4

1. Apply the formula

                                                          a=3, b=x^2, n=4

                                           (3+x^2)^4=\sum_{i=0}^4\binom{4}{i} 3^(4-i) (x^2)^i

2. Expand the summation and simplify

            \sum_{i=0}^4\binom{4}{i} 3^(4-i) (x^2 )^i
                                                                          =\binom{4}{0} 3^4∙(x^2)^0+\binom{4}{1} 3^3∙(x^2 )^1+\binom{4}{2} 3^2∙(x^2)^2+\binom{4}{3} 3^1 ∙(x^2 )^3+\binom{4}{4} 3^0∙(x^2 )^4

Binomial expansions require practice to get the hang of things and to help memorize the formula. If you are interested in more practice problems on this topic or help, check out Symbolab’s Practice.

Until next time,



  1. Good post! You have good mathematical skills and you did a good job by sharing your knowledge with other to help them in their mathematical problem. Keep it up!

  2. This comment has been removed by the author.

  3. Awesome blog. I would love to see true life prepared to walk, so please share more informative updates. Great work keeps it up. Real MB2-715 exam dumps

  4. pasang yuk asianbookie asian games 2018

    pasang yuk judi online asian games

    pasang yuk asianbandar asian games

    pasaran asianbookie asian games 2018 yuk pasang taruhan judi online asian games 2018
    whatup : 6281377055002

  5. Jangan lupa main di Sabung Ayam Online

  6. Here is a list of the games that are unblocked games which you can play at school or work anywhere and anytime ! ✓ No registration required ✓ Totally free ✓

  7. It used to be too hard when I was in School.
    Road Bikes - Back Acne

  8. I think that this website will give you some more useful educative information.

  9. Some other useful information like presentation topics examples you can find here:

  10. mencoba berbagai permainan sabung ayam pw 2019

    Agen Sabung Ayam Online & Agen Sabung Ayam S128
    sabung ayam s128

    Agen Sabung Ayam Online

    produced by s128

  11. You are very good at that.
    โปรโมชั่นGclub ของทางทีมงานตอนนี้แจกฟรีโบนัส 50%
    เพียงแค่คุณสมัคร สล็อตออนไลน์ กับทางทีมงานของเราเพียงเท่านั้น
    สมัครสล็อตออนไลน์ >>> Goldenslot
    สนใจร่วมสนุกกับ คาสิโนออนไลน์ คลิ๊กได้เลย
    มีทั้งคาสิโนออนไลน์ หวยออนไลน์ ฟุตบอลออนไลน์ สล็อตออนไลน์ และอื่นๆอีกมากมาย

  12. There are certainly a lot of details like that to take into consideration. That is a great point to bring up. I offer the thoughts above as general inspiration but clearly there are questions like the one you bring up where the most important thing will be working in honest good faith. I don?t know if best practices have emerged around things like that, but I am sure that your job is clearly identified as a fair game. Both boys and girls feel the impact of just a moment’s pleasure, for the rest of their lives.

    Click Here

  13. We are the best writing company providing Essay Writer Servicesof the highest quality. We guarantee our customers of nothing but the highest quality document. Our Custom Term Paper Writing Services are well referenced, formatted and edited to ensure that the article is rid of unnecessary details.

  14. I love what you guys tend to be up too. This type of clever work and reporting!
    Keep up the amazing works guys I’ve included you guys to our blogroll.

    gmail login

  15. Good post! You have good mathematical skills and you did a good job by sharing your knowledge with other to help them in their mathematical problem. Keep it up!
    Hotspot VPN for PC

  16. It's always good to hear, "buy one and get one free". But with Seers Cookie Consent you can get many more. I will gladly suggest Seers for its tools to generate a complaint Cookie Policy. Do check out its Cookie Consent Banner; it indicates the level of concern Seers possess for its customer's data.
    ccpa compliance
    PECR Cookie Consent

    PECR Cookie Requirement

    GDPR E training

    data protection impact assessment

    gdpr cookie notice

    cookie consent popup

    gdpr cookie banner

  17. The Institute of Data Protection (IDP) is one of the most forward thinking and advanced learning programmes available for data protection professionals. We represent and support our members, promoting the highest professional standards around data protection and privacy issues.
    Institute of Data Protection

  18. Great article. This blog is filled with unique and high-quality articles.
    VPN Master for PC

  19. Yowhatsapp 2020 is getting recognition day by day. People are very dependent on different apps like Facebook, Instagram, WhatsApp, etc. all these apps use for entertaining and communication.

    yowhatsapp 2020

  20. In this game the player have to play as a shark that eats the other fishes to get some treasurers & goes under the water & swim & search for its PREY to hunt it down. By killing the other fishes & getting the treasure there are some achievements / targets in the game that gets unlocked when you achieve them.There are some dangerous fishes also available in the water that could also harm you are JELLY FISHES. Once someone starts playing this game gets addictive to it, because its very entertaining & fun to play game.

    Hungry Shark Evolution

  21. GOGO Live is a live streaming app that allows its members to communicate with the people all around the world. This is such a platform where you get familiar with one another & start to know each other better.

    GOGO Live is a live streaming

  22. Attitude Status for Whatsapp in English: We all have attitude, which we want to express from our Whatsapp Status. So, this time we have collected and updated ...

    attitude status

  23. Bruce Lee Dragon Run for Android is a runner with isometric graphics

  24. Thanks a lot!
    I have read out a numbers of website but could not get complete information only and only this website is complete how to write a proper blog post.

    Coursework writing service

    Coursework writing help

    Coursework on resume

    Do my coursework

  25. Government of Uttar Pradesh has released the list of UP rations card  procedure to check NFSA eligibility list from the online portal of Dept

    uttar pradesh ration card online apply

    ration card online apply up 2019

    ration card check

    rasan card apply

  26. There is a law introduced in UK, that every website that is serving in UK have to place a Cookie Consent Banner on their website to be compliant according to the UK's law