Worksheet: Pascal's Triangle and the Binomial Theorem

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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 Pascal’s triangle and the binomial expansion. He has noticed that each row of Pascal’s triangle can be used to determine the coefficients of the binomial expansion of , as shown in the figure. For example, the fifth row of Pascal’s triangle can be used to determine the coefficients of the expansion of .

By calculating the next row of Pascal’s triangle, find the coefficients of the expansion of .

A1, 7, 21, 34, 34, 21, 7, 1
B1, 7, 21, 35, 35, 21, 7, 1
C1, 6, 8, 20, 8, 1
D2, 8, 12, 8, 2
E1, 6, 15, 20, 15, 6, 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.

A16, 32, 24, 8, 1
B8, 32, 24, 8, 1
C16, 64, 24, 8, 1
D8, 64, 24, 8, 1
E8, 32, 24, 8, 1

Q2:

Find the coefficient of in the expansion of .

Q3:

Matthew knows that he can use the row of Pascal’s triangle to calculate the coefficients of the expansion .

Calculate the numbers in the row of Pascal’s triangle and, hence, write out the coefficients of the expansion .

A1, 6, 15, 20, 15, 6, 1
B1, 5, 10, 10, 5, 1
C1, 7, 21, 35, 35, 21, 7, 1
D2, 6, 15, 20, 15, 6, 2
E1, 3, 3, 1

Now, by considering the different powers of and and using Pascal’s triangle, work out the coefficients of the expansion .

A32, , 320, , 160,
B , 160, , 320, , 32
C32, 160, 320, 320, 160, 32
D64, 160, 320, 320, 160, 64
E64, , 320, , 160,

Q4:

Shown is a partially filled-in picture of Pascal’s 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 .

Q6:

Find the coefficient of in the expansion of .

Q7:

Use Pascal’s triangle to expand the expression .

Q8:

Use Pascal’s triangle to expand the expression .

Q9:

Use Pascal’s triangle to expand the expression .

Q10:

Write the first 5 terms of the expansion of in ascending powers of .

Q11:

In the expansion of a binomial, determine which of the following is equivalent to the relation .

Q12:

Use Pascal’s triangle to determine the coefficients of the terms that result from the expansion of .