Math: Pascal's Triangle and the Binomial Theorem
Expanding binomials and finding coefficients
Math: Pascal's Triangle and the Binomial Theorem
Expanding binomials and finding coefficients
Math - Grade 9-12
- 1
Write rows 0 through 5 of Pascal's Triangle. Remember that row 0 is the single number 1.
Each interior number is the sum of the two numbers directly above it.
Rows 0 through 5 are: row 0 is 1; row 1 is 1, 1; row 2 is 1, 2, 1; row 3 is 1, 3, 3, 1; row 4 is 1, 4, 6, 4, 1; row 5 is 1, 5, 10, 10, 5, 1. - 2
Use Pascal's Triangle to expand (x + y)^4.
The expansion is x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4. The coefficients come from row 4 of Pascal's Triangle. - 3
Use Pascal's Triangle to expand (a + b)^5.
Use the row 1, 5, 10, 10, 5, 1.
The expansion is a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5. The coefficients come from row 5 of Pascal's Triangle. - 4
Expand (2x + 3)^3 completely.
The expansion is 8x^3 + 36x^2 + 54x + 27. This comes from (2x)^3 + 3(2x)^2(3) + 3(2x)(3)^2 + 3^3. - 5
Find the coefficient of x^3 in the expansion of (x + 2)^5.
To get x^3, choose the term where the power of x is 3 and the power of 2 is 2.
The coefficient of x^3 is 40. The x^3 term is C(5, 2)x^3(2)^2, which equals 10 times 4 times x^3, so the coefficient is 40. - 6
Use the Binomial Theorem to find the fourth term of (3x - 2)^6 when written in descending powers of x.
The fourth term is -4320x^3. The fourth term has k = 3, so it is C(6, 3)(3x)^3(-2)^3 = 20 times 27x^3 times -8 = -4320x^3. - 7
Find the missing entries in this row of Pascal's Triangle: 1, 7, __, __, __, __, 7, 1.
This is row 7, so the entries are C(7, 0), C(7, 1), C(7, 2), and so on.
The complete row is 1, 7, 21, 35, 35, 21, 7, 1. Each interior number can be found by adding the two numbers above it, or by using combinations from row 7. - 8
Explain why the coefficients in the expansion of (x + y)^n add up to 2^n.
Think about what happens if both variables are replaced by 1.
The coefficients add up to 2^n because substituting x = 1 and y = 1 gives (1 + 1)^n = 2^n. The expansion then becomes the sum of all the coefficients. - 9
Find the sum of the coefficients in the expansion of (2x - 5)^8.
The sum of the coefficients is 6561. To find it, substitute x = 1, giving (2(1) - 5)^8 = (-3)^8 = 6561. - 10
Find the coefficient of x^4 in the expansion of (x - 3)^7.
The exponent on x is 7 - k, so set 7 - k = 4.
The coefficient of x^4 is -945. The x^4 term occurs when k = 3, so the term is C(7, 3)x^4(-3)^3 = 35 times -27 times x^4 = -945x^4. - 11
A student claims that the middle number in row 8 of Pascal's Triangle is 56. Check the claim and explain whether it is correct.
The claim is not correct. Row 8 is 1, 8, 28, 56, 70, 56, 28, 8, 1, so the middle number is 70. - 12
Use the Binomial Theorem to write the general term of (a + b)^n, then use it to identify the term containing a^2b^5 in (a + b)^7.
Match the exponent of b to k and the exponent of a to n - k.
The general term is C(n, k)a^(n - k)b^k. In (a + b)^7, the term containing a^2b^5 has k = 5, so it is C(7, 5)a^2b^5 = 21a^2b^5.