4.1.1

Pascal's Triangle

Test yourself

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(x+y)^n.

Illustrative background for How do you construct Pascal's triangle?Illustrative background for How do you construct Pascal's triangle? ?? "content

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.
Illustrative background for Pascal's triangleIllustrative background for Pascal's triangle ?? "content

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.
Illustrative background for Expanding binomialsIllustrative background for Expanding binomials ?? "content

Expanding binomials

  • We can use the numbers in each row of a pascal triangle to expand expressions of the form (x+y)n(x+y)^n.
  • The first row is equal to (x+y)0(x+y)^0.
Illustrative background for Expanding binomialsIllustrative background for Expanding binomials ?? "content

Expanding binomials

  • The second row gives the coefficients of the terms of the (x+y)1(x+y)^1 expansion.
  • The coefficients of the terms of the (x+y)n(x+y)^{n} expansion are given by the (n+1)(n+1)th row of Pascal's triangle.
Illustrative background for Expanding binomialsIllustrative background for Expanding binomials ?? "content

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=x3+3x2y1+3y2x1+y3(x+y)^3 = x^3 + 3x^2y^1+3y^2x^1 +y^3
    • Each term contains powers that add up to three.
Illustrative background for ExampleIllustrative background for Example ?? "content

Example

  • What is the expansion of (2x+3)4(2x+3)^4?
Illustrative background for Write out Pascal's triangle up to row 5Illustrative background for Write out Pascal's triangle up to row 5 ?? "content

Write out Pascal's triangle up to row 5

  • The coefficients of the terms in the expansion (2x+3)4(2x+3)^4 are given by the 55th row of Pascal's triangle.
Illustrative background for Write out coefficients of expansionIllustrative background for Write out coefficients of expansion ?? "content

Write out coefficients of expansion

  • The coefficients are:
    • 1,4,6,4,11,4,6,4,1
Illustrative background for Multiply coefficientsIllustrative background for Multiply coefficients ?? "content

Multiply coefficients

  • Multiply each coefficient by decreasing orders of 2x2x and increasing orders of 33:
    • (2x)430(2x)331(2x)232(2x)132(2x)034(2x)^43^0\rightarrow(2x)^33^1\rightarrow(2x)^23^2\rightarrow(2x)^13^2\rightarrow(2x)^03^4
Illustrative background for Write the sum of the termsIllustrative background for Write the sum of the terms ?? "content

Write the sum of the terms

  • The answer is the sum of each term:
    • (2x+3)4=(2x)4+4(2x)331+6(2x)232+4(2x)233+34(2x+3)^4 = (2x)^4 + 4(2x)^33^1 + 6(2x)^23^2+4(2x)^23^3 + 3^4
Illustrative background for AnswerIllustrative background for Answer ?? "content

Answer

  • We can simplify our answer by working out the powers:
    • (2x+3)4=16x4+96x3+216x2+216x+81(2x+3)^4 = 16x^4+96x^3 + 216x^2 +216x + 81

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

Go student ad image

Unlock your full potential with GoStudent tutoring

  • Affordable 1:1 tutoring from the comfort of your home

  • Tutors are matched to your specific learning needs

  • 30+ school subjects covered

Book a free trial lesson