• Theory and examples
• Exercises

### 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:

${{(a+b)}^{2}}={{a}^{2}}+2ab+{{b}^{2}} \quad \quad (1)$

There are three other binomial formulas:

${{(a-b)}^{2}}={{a}^{2}}-2ab+{{b}^{2}} \quad \quad (2)$

$(a+b)(a-b)={{a}^{2}}-{{b}^{2}} \quad \quad (3)$

$(a-b)(a+b)={{a}^{2}}-{{b}^{2}} \quad \quad (4)$

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:

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:

$(a+b)(c+d)=a\cdot c+a\cdot d+b\cdot c+b\cdot d \quad \quad (1')$

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

$(a+b)(c-d)=a\cdot c-a\cdot d+b\cdot c-b\cdot d \quad \quad (2')$

$(a-b)(c+d)=a\cdot c+a\cdot d-b\cdot c-b\cdot d \quad \quad (3')$

$(a-b)(c-d)=a\cdot c-a\cdot d-b\cdot c+b\cdot d \quad \quad (4')$

1)

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

a)

$(3+x)^2$

b)

$(8a-2b)^2$

c)

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

2)

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

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)

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

Further exercises and tutorials can be found on: