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Binomial Expansion

How to use Pascal’s triangle to expand powers of brackets and find selected coefficients without expanding everything.

Use the correct row of Pascal’s triangle

For a positive integer n, the expansion of (a+b)^n uses the coefficients in row n of Pascal’s triangle. The top 1 is row 0. Each new row starts and ends with 1; an inside entry is the sum of the two above it.

For power 3, use 1,3,3,1:

a^3+3a^2b+3ab^2+b^3

The power of a goes down while the power of b goes up. Their sum is always 3

Explore the coefficients

Try it: compare powers 3, 4 and 5. There are n+1 coefficients in row n, and they read the same from either end.

Worked example 1

Expand (x+4)^3

  1. Use x^3+3x^2(4)+3x(4^2)+4^3

  2. Your turn. What is the coefficient of x, 3\times4^2?

    The coefficient of x is 48

  3. Answer: x^3+12x^2+48x+64

Raise the whole term to its power

In (2x-3)^3, use a=2x and b=-3. Thus a^3=(2x)^3=8x^3. A negative second term gives alternating signs because its odd powers are negative.

Worked example 2

Expand (2x-3)^3

  1. The four terms are (2x)^3, 3(2x)^2(-3), 3(2x)(-3)^2 and (-3)^3

  2. Your turn. Work out the coefficient of x^2: 3\times4\times(-3)

    The coefficient of x^2 is -36

  3. Answer: 8x^3-36x^2+54x-27

Find only the term you need

To find the coefficient of x^r in (a+bx)^n, select the Pascal entry for the term containing (bx)^r, then multiply by a^{n-r}b^r. Count entries starting at r=0. You usually do not need the entire expansion.

Worked example 3

Find the coefficient of x^2 in (1+3x)^5

  1. Row 5 is 1,5,10,10,5,1. The x^2 term uses the third entry, 10

  2. Your turn. Work out 10\times3^2

    10(3x)^2=90x^2

  3. Answer: 90. The coefficient is the number multiplying x^2, not the whole term.

Worked example 4

Find the coefficient of x^3 in (2-x)^6

  1. Row 6 is 1,6,15,20,15,6,1. Select 20(2^3)(-x)^3

  2. Your turn. Work out 20\times8\times(-1)

    The term is -160x^3

  3. Answer: -160. The factor 2^3 and the negative sign are both essential.

Worked example 5

The coefficient of x^2 in (1+kx)^4 is 150. Find all possible real values of k.

  1. Row 4 gives the x^2 term 6(kx)^2=6k^2x^2. Therefore 6k^2=150

  2. Your turn. What is k^2?

    k^2=25

  3. Answer: k=5 or k=-5. The even power means both signs give the same x^2 coefficient.

Worked example 6

Find the coefficient of x^3 in (1+2x)(3-x)^4

  1. There are two contributions: 1 times the x^3 term of (3-x)^4, and 2x times its x^2 term.

  2. Your turn. Find the x^2 coefficient in (3-x)^4: 6\times3^2\times(-1)^2

    The x^2 term is 54x^2

  3. The x^3 coefficient is 4\times3\times(-1)^3=-12
  4. Answer: -12+2(54)=96. Both contributions have total power 3

Common mistakes

  • Using the wrong Pascal row. For power n, use row n, counting the top 1 as row 0.
  • Forgetting powers of coefficients. Raise the whole term, including its numerical coefficient, to the power.
  • Losing signs of powers. An odd power of a negative term stays negative; an even power is positive.
  • Giving a term instead of its coefficient. If only the coefficient is requested, give the multiplying number.
  • Missing contributions to a power. Find every pair of terms whose powers add to the required power.

Now try it: Binomial Expansion practice questions

More on this topic: Binomial Expansion worksheet with full solutions