Worksheet: Pascal’s Triangle and the Binomial Theorem
In this worksheet, we will practice using Pascal’s triangle to find the coefficients of the algebraic expansion of any binomial expression of the form (a+b)ⁿ.
Q1:
Daniel has been exploring the relationship between Pascalโs triangle and the binomial expansion. He has noticed that each row of Pascalโs triangle can be used to determine the coefficients of the binomial expansion of , as shown in the figure. For example, the fifth row of Pascalโs triangle can be used to determine the coefficients of the expansion of .
By calculating the next row of Pascalโs triangle, find the coefficients of the expansion of .
- A1, 7, 21, 35, 35, 21, 7, 1
- B2, 8, 12, 8, 2
- C1, 7, 21, 34, 34, 21, 7, 1
- D1, 6, 15, 20, 15, 6, 1
- E1, 6, 8, 20, 8, 1
Daniel now wants to calculate the coefficients for each of the terms of the expansion . By substituting into the expression above, or otherwise, calculate all of the coefficients of the expansion.
- A8, 64, 24, 8, 1
- B8, 32, 24, 8, 1
- C16, 64, 24, 8, 1
- D16, 32, 24, 8, 1
- E8, 32, 24, 8, 1
Q3:
Matthew knows that he can use the row of Pascalโs triangle to calculate the coefficients of the expansion .
Calculate the numbers in the row of Pascalโs triangle and, hence, write out the coefficients of the expansion .
- A1, 6, 15, 20, 15, 6, 1
- B2, 6, 15, 20, 15, 6, 2
- C1, 7, 21, 35, 35, 21, 7, 1
- D1, 5, 10, 10, 5, 1
- E1, 3, 3, 1
Now, by considering the different powers of and and using Pascalโs triangle, work out the coefficients of the expansion .
- A32, , 320, , 160,
- B64, 160, 320, 320, 160, 64
- C32, 160, 320, 320, 160, 32
- D64, , 320, , 160,
- E, 160, , 320, , 32
Q4:
Shown is a partially filled-in picture of Pascalโs triangle. By spotting patterns, or otherwise, find the values of , , , and .
- A, , , and
- B, , , and
- C, , , and
- D, , , and
- E, , , and
Q5:
Fully expand the expression .
- A
- B
- C
- D
- E
Q6:
Find the coefficient of in the expansion of .
Q7:
Use Pascalโs triangle to expand the expression .
- A
- B
- C
- D
- E
Q10:
Write the first 5 terms of the expansion of in ascending powers of .
- A
- B
- C
- D
- E
Q11:
In the expansion of a binomial, determine which of the following is equivalent to the relation .
- A
- B
- C
- D
Q12:
Use Pascalโs triangle to determine the coefficients of the terms that result from the expansion of .
- A
- B
- C
- D
- E
Q14:
Find the coefficient of in the expansion of .
- A46,189,440
- B
- C
- D
- E
Q16:
Write the coefficients of the terms that result from the expansion of .
- A
- B
- C
- D
- E
Q17:
Find the product of the coefficients of the terms of the expansion of .