How do I use Pascal's triangle to expand (x - 1)^5?

Aug 19, 2014

The answer is: ${x}^{5} - 5 {x}^{4} + 10 {x}^{3} - 10 {x}^{2} + 5 x - 1$

When expanding, we consider the general form: ${\left(x + y\right)}^{n}$.

Recall that the first row of Pascal's Triangle is: ${\left(x + y\right)}^{0}$. So for ${\left(x - 1\right)}^{5}$, we are looking at the ${6}^{t h}$ row of Pascal's Triangle for the coefficients:

color(white)((color(black)((,,,,,1,,,,,),(,,,,1,,1,,,,),(,,,1,,2,,1,,,),(,,1,,3,,3,,1,,),(,1,,4,,6,,4,,1,),(color(red)1,,color(blue)5,,color(green)10,,color(orange)10,,color(olive)5,,color(pink)1)))

Expanding, we get:

$\textcolor{red}{1} \cdot {x}^{5} {y}^{0} + \textcolor{b l u e}{5} \cdot {x}^{4} {y}^{1} + \textcolor{g r e e n}{10} \cdot {x}^{3} {y}^{2} + \textcolor{\mathmr{and} a n \ge}{10} \cdot {x}^{2} {y}^{3} + \textcolor{o l i v e}{5} \cdot {x}^{1} {y}^{4} + \textcolor{\pi n k}{1} \cdot {x}^{0} {y}^{5}$

Now we substitute and simplify:

${x}^{5} + 5 {x}^{4} {\left(- 1\right)}^{1} + 10 {\cdot}^{3} {\left(- 1\right)}^{2} + 10 {x}^{2} {\left(- 1\right)}^{3} + 5 {x}^{1} {\left(- 1\right)}^{4} + {\left(- 1\right)}^{5}$
$= {x}^{5} - 5 {x}^{4} + 10 {x}^{3} - 10 {x}^{2} + 5 x - 1$