Algebraic Binomial Operation

Binomial Theorem

One of the most common operation in Algebra is binomial operation. The theorem above shows how it goes. But there's its operational which is easy to understand and to use. Below is the general form of Algebraic Binomial expansion operation. The "x" and "y" were colored to easily spot the behavior when they are in the equation. See below;

$({\color{Blue} x}+{\color{Red} y})^{n}={\color{Blue} x}^{n}+n{\color{Blue} x}^{n-1}{\color{Red} y}+\frac{n(n-1)}{2!}{\color{Blue} x}^{n-2}{\color{Red}y}^{2}+...+{\color{Red} y}^{n}$

Moreover, additional 'per piece' operation were provided below.

Let's go to Illustrative examples:

Illustrative Examples:

$({\color{Red} a}+{\color{DarkGreen} b})^{0}=1$

$({\color{Red} a}+{\color{DarkGreen} b})^{1}={\color{Red} a}+{\color{DarkGreen} b}$

$({\color{Red} a}+{\color{DarkGreen} b})^{2}={\color{Red} a}^{2}+2{\color{Red} a}{\color{DarkGreen} b}+{\color{DarkGreen}b}^{2}$

$({\color{Red} a} + {\color{DarkGreen} b})^{3} = ({\color{Red} a} + {\color{DarkGreen} b})({\color{Red} a} + {\color{DarkGreen} b})^{^{2}} = ({\color{Red} a} + {\color{DarkGreen} b})({\color{Red} a}^{2} + 2{\color{Red} a}{\color{DarkGreen} b} + {\color{DarkGreen} b}^{2}) = {\color{Red} a}^{^{3}} + 3{\color{Red} a}^{2}{\color{DarkGreen} b} + 3{\color{Red} a}{\color{DarkGreen} b}^{2}+ {\color{DarkGreen} b}^{3}$

Usually, binomial to up to the third power is given in every classroom discusion. Higher than that will be discussed in another page.

Example Problems:

Expand the given binomials:

$1.) ({\color{Red} x} + 2{\color{DarkGreen} y})^{2} =$

$({\color{Red} x} + 2{\color{DarkGreen} y})^{2} = {\color{Red} x}^{2} + 2({\color{Red} x})(2{\color{DarkGreen} y}) + (2{\color{DarkGreen} y})^{2}$

$\({\color{Red} x} + 2{\color{DarkGreen} y})^{2} = {\color{Red} x}^{2} + 4{\color{Red} x}{\color{DarkGreen} y} + 4{\color{DarkGreen} y}^{2}$ , final answer

$2.) ({\color{Red} a}+{\color{DarkGreen} 3})^{3} =$

$({\color{Red} a}+{\color{DarkGreen} 3})^{3} = ({\color{Red} a}+{\color{DarkGreen} 3})({\color{Red} a}+{\color{DarkGreen} 3})^{2}$

$({\color{Red} a}+{\color{DarkGreen} 3})^{3} =({\color{Red} a}+{\color{DarkGreen} 3}) ({\color{Red} a}^{2} + 2{\color{Red} a}({\color{DarkGreen} 3}) +{\color{DarkGreen} 3}^{2})$

$({\color{Red} a}+{\color{DarkGreen} 3})^{3} =({\color{Red} a}+{\color{DarkGreen} 3})({\color{Red} a}^{2} +6{\color{Red} a} +{\color{DarkGreen} 3}^{2})$

$({\color{Red} a}+{\color{DarkGreen} 3})^{3} =({\color{Red} a}^{3} +{\color{Red} a}^{2}6 +9{\color{Red} a} +3{\color{Red} a}^{2} +18{\color{Red} a} +{\color{DarkGreen} 3}^{3})$ , collect similar terms

$({\color{Red} a}+{\color{DarkGreen} 3})^{3} ={\color{Red} a}^{3} + 9{\color{Red} a}^{2} +27{\color{Red} a} +27$ , final answer

$3.) ({\color{Red} x} - {\color{DarkGreen} 2})^{3}=$

$({\color{Red} x} - {\color{DarkGreen} 2})^{3}=({\color{Red} x} - {\color{DarkGreen} 2})({\color{Red} x} - {\color{DarkGreen} 2})^{3}$

$({\color{Red} x} - {\color{DarkGreen} 2})^{3}=({\color{Red} x} - {\color{DarkGreen} 2})({\color{Red} x}^{2}-{\color{DarkGreen} 2}{\color{Red} x} +{\color{DarkGreen} 2}^{2})$

$({\color{Red} x} - {\color{DarkGreen} 2})^{3}={\color{Red} x}^{3} -2{\color{Red} x}^{2} +4{\color{Red} x} -2{\color{Red} x}^{2} +4{\color{Red} x} -{\color{DarkGreen} 2}^{3}$

$({\color{Red} x} - {\color{DarkGreen} 2})^{3}={\color{Red} x}^{3} -4{\color{Red} x}^{2} +8{\color{Red} x} -8$ , final answer

$4.) ({\color{Red} 2a} -{\color{DarkGreen} b})^{4}=$

$({\color{Red} 2a} -{\color{DarkGreen} b})^{4}=({\color{Red} 2a} -{\color{DarkGreen} b})^{2}({\color{Red} 2a} -{\color{DarkGreen} b})^{2}$

$({\color{Red} 2a} -{\color{DarkGreen} b})^{4}=[({\color{Red} 2a})^{2}-2({\color{Red} 2a}){\color{DarkGreen} b} +{\color{DarkGreen} b}^{2}][({\color{Red} 2a})^{2}-2({\color{Red} 2a}){\color{DarkGreen} b} +{\color{DarkGreen} b}^{2}]$

$({\color{Red} 2a} -{\color{DarkGreen} b})^{4}=[({\color{Red} 2a})^{4}-2({\color{Red} 2a})^{3}{\color{DarkGreen} b} +({\color{Red} 2a})^{2}{\color{DarkGreen} b}^{2} -2({\color{Red} 2a})^{3}{\color{DarkGreen} b} +4({\color{Red} 2a})^{2}{\color{DarkGreen} b}^{2} -2({\color{Red} 2a}){\color{DarkGreen} b}^{3} +({\color{Red} 2a})^{2}{\color{DarkGreen} b}^{2} -2({\color{Red} 2a}) {\color{DarkGreen} b}^{3} +{\color{DarkGreen} b}^{4}]$

$({\color{Red} 2a} -{\color{DarkGreen} b})^{4}=16{\color{Red} a}^{4}-32{\color{Red} a}^{3}{\color{DarkGreen} b }+24{\color{Red} a}^{2}{\color{DarkGreen} b}^{2} -20{\color{Red} a}{\color{DarkGreen} b}^{3} +{\color{DarkGreen} b}^{4}$ , final answer

$5.)({\color{Red} a} + 2{\color{DarkGreen} b})^{5}=$

$({\color{Red} a} + 2{\color{DarkGreen} b})^{5}={\color{Red} a}^{5} + 5{\color{Red} a}^{4}(2{\color{DarkGreen} b}) + 10{\color{Red}a}^{3}(2{\color{DarkGreen} b})^{2} + 10{\color{Red} a}^{2}(2b)^{3} + 5{\color{Red} a}(2{\color{DarkGreen} b})^{4 }+ 1(2{\color{DarkGreen} b})^{5}$

$({\color{Red} a} + 2{\color{DarkGreen} b})^{5}= {\color{Red} a}^{5} + 10{\color{Red} a}4{\color{DarkGreen} b} + 40{\color{Red} a}3{\color{DarkGreen} b}^{2} + 80{\color{Red} a}2{\color{DarkGreen} b}^{3} + 80{\color{Red} a}{\color{DarkGreen} b}^{4 }+ 32{\color{DarkGreen} b}^{5}$ , final answer

Sample Quiz:

Expand the following binomials:

$1. ) (a + 5)^{2} =$

$(a + 5)^{2} =[a^{2}+(2)5a +5^{2})]$

$(a + 5)^{2} =a^{2}+({\color{Red} supply}) +5^{2})$ , click here to write the middle term

$({\color{Red} a} + {\color{DarkGreen} 5})^{2} ={\color{Red} a}^{2}+10{\color{Red} a} +{\color{DarkGreen} 25}$ , is this the right answer? Leave a comment here.

$2.) (2a+3)^{2}=$

   $(2a+3)^{2}= [{({\color{Red} supply})} +2(2a)3 +3^{2}]$ , click here to write the first term

   $(2{\color{Red} a}+3)^{2}= 4{\color{Red} a}^{2} +12{\color{Red} a} +3^{2}$ , is this the right answer? Leave a comment here.

$3.) (3 -b)^{2}=$

   $(3 -b)^{2}= [3^{2}-2(3)b +({\color{Red} supply})]$ , click here to write the last term

   $({\color{Red} 3} -{\color{Blue} b)}^{2}= {\color{Red} 3}^{2} -6{\color{Blue} b} +{\color{DarkBlue} b}^{2}$ , is this the right answer? Leave a comment here.

POP QUIZ:

Expand the given binomials:

$1. ) (a+1)^{2}=$ "click here"

$2. ) (2x+y)^{2}=$ "click here"

$3. ) (3a-2b)^{2}=$ "click here"

$4. ) (a-2b)^{3}=$ "click here"

$5. ) (a-2)^{3}=$ "click here"

"If you see typographical or any error(s), please feel free "Leave a comment" below."

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DMG

Algebraic Binomial Operation

Posted by : Allan_Dell on Monday, April 6, 2020 | 3:45 PM

Popular posts

DMG

Algebraic Binomial Operation

Posted by : Allan_Dell on Monday, April 6, 2020 | 3:45 PM

Popular posts

Posted by : Allan_Dell on Monday, April 6, 2020 | 3:45 PM