### Video Transcript

Use Pascalβs triangle to expand the
expression three plus π₯ to the power of four.

Pascalβs triangle can be used to
determine the expanded pattern of coefficients. As the name suggests, it is
triangular in shape. We can find the next row of the
triangle by adding the two numbers above as shown in the diagram. The power, or exponent, in our
question is four. This means that there will be five
terms in the expansion of the parentheses. The key row is one, four, six,
four, and one. These will be the coefficients of
each of the five terms of our expansion.

We could also calculate these
values by choosing the nCr button on the calculator. Four choose zero is equal to one,
four choose one is equal to four, four choose two is equal to six, and so on. The rest of the expression will be
π to the power of four multiplied by π to the power of zero, π cubed multiplied
by π to the power of one, π squared multiplied by π squared, and so on. The powers of π decrease from four
to zero, and the powers of π increase from zero to four, where π is the first term
in the parentheses, and π is the second.

This expansion works for any
expression of the form π plus π to the power of π, where π is a positive
integer. Substituting in our values of π
and π gives the following. We know that anything to the power
of zero is equal to one. Therefore, π₯ to the power of zero
and three to the power of zero are equal to one. Three squared is equal to nine. Three cubed is equal to 27. And three to the power of four is
equal to 81.

The first term simplifies to one
multiplied by 81. The second term simplifies to four
multiplied by 27 multiplied by π₯. The third term is six multiplied by
nine multiplied by π₯ squared. The fourth term is four multiplied
by three multiplied by π₯ cubed. The final term is one π₯ to the
power of four. Simplifying each of the terms gives
us 81 plus 108π₯ plus 54π₯ squared plus 12π₯ cubed plus π₯ to the power of four. This is the binomial expansion of
three plus π₯ to the power of four using Pascalβs triangle.