# Powers of the Binomial

## Key Questions

The power of a binomial is the value of $n$ in the binomial expression ${\left(a + x\right)}^{n}$.

#### Explanation:

For any value of $n$, the ${n}^{\text{th}}$ power of a binomial is given by:

(x+y)^n=x^n +nx^(n-1)y +(n(n-1))/2x^(n-2)y^2 + … + y^n

The general formula for the expansion is:

(x+y)^n = sum_(k=0)^n (n!)/((n-k)!k!)x^(n-k)y^k

The coefficients for varying $x$ and $y$ can be arranged to form Pascal's triangle.

The ${n}^{\text{th}}$ row in the triangle gives the coefficients of the terms in the ${\left(n - 1\right)}^{\text{th}}$ power of the polynomial.

${\left(a + b\right)}^{3} = {a}^{3} + 3 {a}^{2} b + 3 a {b}^{2} + {b}^{3}$

#### Explanation:

The coefficients $1 , 3 , 3 , 1$ can be found as a row of Pascal's triangle:

For other powers of a binomial use a different row of Pascal's triangle.

For example:

${\left(a + b\right)}^{5} = {a}^{5} + 5 {a}^{4} b + 10 {a}^{3} {b}^{2} + 10 {a}^{2} {b}^{3} + 5 a {b}^{4} + {b}^{5}$

How about ${\left(2 x - 5\right)}^{3}$ or similar?

Let $a = 2 x$ and $b = - 5$ to find:

${\left(2 x - 5\right)}^{3}$

$= {\left(a + b\right)}^{3} = {a}^{3} + 3 {a}^{2} b + 3 a {b}^{2} + {b}^{3}$

$= {\left(2 x\right)}^{3} + 3 {\left(2 x\right)}^{2} \left(- 5\right) + 3 \left(2 x\right) {\left(- 5\right)}^{2} + {\left(- 5\right)}^{3}$

$= 8 {x}^{3} - 60 {x}^{2} + 150 x - 125$

I am not sure about what you need, but have a look:

#### Explanation:

If you have a binomial such as $\left(a + b\right)$ and you square it you get:
${\left(a + b\right)}^{2}$
but this is the same as:
$\left(a + b\right) \left(a + b\right)$

exactly as ${4}^{2} = 4 \times 4$.

The result of $\left(a + b\right) \left(a + b\right)$ is interesting because you need to multiply each term of the first bracket by each term of the second and add the results!
$\left(a + b\right) \left(a + b\right) = a \cdot a + a \cdot b + b \cdot a + b \cdot b = {a}^{2} + 2 a b + {b}^{2}$

Try by yourself with a difficult one: ${\left(a - b\right)}^{2}$ remembering to consider the signs of each term!