Question Video: Use Pascal’s Triangle to Expand the Expression Mathematics

Video Transcript

Use Pascal’s triangle to expand the expression three plus 𝑥 to the power of four.

Pascal’s 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 𝑎 to the power of four multiplied by 𝑏 to the power of zero, 𝑎 cubed multiplied by 𝑏 to the power of one, 𝑎 squared multiplied by 𝑏 squared, and so on. The powers of 𝑎 decrease from four to zero, and the powers of 𝑏 increase from zero to four, where 𝑎 is the first term in the parentheses, and 𝑏 is the second.

This expansion works for any expression of the form 𝑎 plus 𝑏 to the power of 𝑛, where 𝑛 is a positive integer. Substituting in our values of 𝑎 and 𝑏 gives the following. We know that anything to the power of zero is equal to one. Therefore, 𝑥 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 𝑥. The third term is six multiplied by nine multiplied by 𝑥 squared. The fourth term is four multiplied by three multiplied by 𝑥 cubed. The final term is one 𝑥 to the power of four. Simplifying each of the terms gives us 81 plus 108𝑥 plus 54𝑥 squared plus 12𝑥 cubed plus 𝑥 to the power of four. This is the binomial expansion of three plus 𝑥 to the power of four using Pascal’s triangle.