Preparatory Course

Binomial Formulas

1. Binomial formulas in compact form

The expression \((a+b)^2\) has to be expanded. The formula showing how to do so is called the first binomial formula:

There are three other binomial formulas:

Examples

\[{{(x+3)}^{2}}={{x}^{2}}+6x+9\]

\[{{(4x-y)}^{2}}=16{{x}^{2}}-8xy+{{y}^{2}}\]

\[(2z+5)(2z-5)=4{{z}^{2}}-25\]

Of course, these examples can also be solved by expanding the terms; the knowledge of these formulas is useful and timesaving since these terms appear frequently.

 

2. Derivation

The expression \((a+b)^2\) can be expanded as follows:

\[{{(a+b)}^{2}}=(a+b)(a+b)=a\cdot a+a\cdot b+b\cdot a+b\cdot b={{a}^{2}}+2ab+{{b}^{2}}\]

Two remarks:

• The chapter Multiplication and Expanding shows in detail how a product of two sums is expanded.

• A frequent error consists of neglecting the term 2ab (which is commonly denoted as the "mixed term").

 

Note that \((a+b)^2\) cannot be the same as \(a^2+b^2\). Two illustrations demonstrate this inconsistency:

Numerical example

Graphical example

 

Expanding also shows

\[{{(a-b)}^{2}}=(a-b)(a-b)=a\cdot a-a\cdot b-b\cdot a+b\cdot b={{a}^{2}}-2ab+{{b}^{2}}\]

\[(a+b)(a-b)=a\cdot a-a\cdot b+b\cdot a-b\cdot b={{a}^{2}}-{{b}^{2}}\]

 

3. Generalization

The term \((a+b)^2=(a+b)(a+b)\) is a special case of \((a+b)(c+d)\). In the above-mentioned case, \(a=c\) and \(b=d\). We will now show how to expand the general term \((a+b)(c+d)\) correctly:

Binomials can also be expanded if values inside brackets are subtracted. The remaining three formulas are:

1)

Expand the following terms using the binomial formulas (1-4):

a)

$(3+x)^2$

b)

$(8a-2b)^2$

c)

$(7-x)\cdot(7+x)$

Solution

 

2)

How can you expand the term \((a+b)^3\)?

Solution

 

3)

Expand the following terms using the binomial formulas (1'-4'):

a)

$(2+x) \cdot (y-3)$

b)

$(x-10) \cdot (y-5)$

c)

$(3-a)\cdot(y+b)$

d)

$(3x-y) \cdot (y+2x)$

Solution

 

Further exercises and tutorials can be found on:

Mathsteacher

Mathwarehouse