Pascal's Triangle

Maths: Edexcel A Level Pure Maths, topic 4.1.1

What is Pascal's Triangle?

Pascal's triangle is an arrangement of numbers that allow us to quickly work out the coefficients of terms when you expand expressions of the form $(x+y)^n$ .

How do you construct Pascal's triangle?

How do you construct Pascal's triangle?

To construct Pascal's triangle, we start by writing a 1.
In the row below, we write two 1’s either side.
For the third row, we add the two 1's in the second row to give 2, and write this in between them on the row below.
The start and ending numbers of a row are always equal to 1.

Pascal's triangle

Pascal's triangle

We continue this way, adding two numbers to find the number in the row below it until we have written out as many rows as we would like.

Expanding binomials

Expanding binomials

We can use the numbers in each row of a pascal triangle to expand expressions of the form $(x+y)^n$ .
The first row is equal to $(x+y)^0$ .

Expanding binomials

Expanding binomials

The second row gives the coefficients of the terms of the $(x+y)^1$ expansion.
The coefficients of the terms of the $(x+y)^{n}$ expansion are given by the $(n+1)$ th row of Pascal's triangle.

Expanding binomials

Expanding binomials

Each term in the expansion has a total power that is equal to the power of the expansion.
For example, $(x+y)^3 = x^3 + 3x^2y^1+3y^2x^1 +y^3$
Each term contains powers that add up to three.

Example

Example

What is the expansion of $(2x+3)^4$ ?

Write out Pascal's triangle up to row 5

Write out Pascal's triangle up to row 5

The coefficients of the terms in the expansion $(2x+3)^4$ are given by the $5$ th row of Pascal's triangle.

Write out coefficients of expansion

Write out coefficients of expansion

The coefficients are:
$1,4,6,4,1$

Multiply coefficients

Multiply coefficients

Multiply each coefficient by decreasing orders of $2x$ and increasing orders of $3$ :
$(2x)^43^0\rightarrow(2x)^33^1\rightarrow(2x)^23^2\rightarrow(2x)^13^2\rightarrow(2x)^03^4$

Write the sum of the terms

Write the sum of the terms

The answer is the sum of each term:
$(2x+3)^4 = (2x)^4 + 4(2x)^33^1 + 6(2x)^23^2+4(2x)^23^3 + 3^4$

Answer

Answer

We can simplify our answer by working out the powers:
$(2x+3)^4 = 16x^4+96x^3 + 216x^2 +216x + 81$

Test yourself on Pascal's Triangle

What is the value of A in the Pascal triangle shown?
What is the value of A in the Pascal triangle shown?
What is the value of A in the Pascal triangle shown?

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