MathsAges 16-18

Binomial theorem coefficients

Use combinations C(n, r) and powers to find specific coefficients in the binomial expansion of (x + k)ⁿ. Review suitability for your learners before teaching.

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Worksheet StudioPractice pack · Ages 16-18

Binomial theorem coefficients

Take your time. Show your thinking.

1. Find the coefficient of x^1 in the binomial expansion of (x + 3)^4.
1 · 3 · 4 · Binomial theorem coefficie…
2. Find the coefficient of x^3 in the binomial expansion of (x + 1)^6.
3 · 1 · 6 · Binomial theorem coefficie…
3. Find the coefficient of x^2 in the binomial expansion of (x + 4)^3.
2 · 4 · 3 · Binomial theorem coefficie…
4. Find the coefficient of x^3 in the binomial expansion of (x + 3)^6.
3 · 3 · 6 · Binomial theorem coefficie…
5. Find the coefficient of x^5 in the binomial expansion of (x + 2)^6.
5 · 2 · 6 · Binomial theorem coefficie…
6. Find the coefficient of x^1 in the binomial expansion of (x + 4)^6.
1 · 4 · 6 · Binomial theorem coefficie…
7. Find the coefficient of x^1 in the binomial expansion of (x + 4)^4.
1 · 4 · 4 · Binomial theorem coefficie…
8. Find the coefficient of x^4 in the binomial expansion of (x + 7)^5.
4 · 7 · 5 · Binomial theorem coefficie…
9. Find the coefficient of x^1 in the binomial expansion of (x + 6)^4.
1 · 6 · 4 · Binomial theorem coefficie…
10. Find the coefficient of x^4 in the binomial expansion of (x + 1)^6.
4 · 1 · 6 · Binomial theorem coefficie…
11. Find the coefficient of x^1 in the binomial expansion of (x + 5)^4.
1 · 5 · 4 · Binomial theorem coefficie…
12. Find the coefficient of x^3 in the binomial expansion of (x + 5)^4.
3 · 5 · 4 · Binomial theorem coefficie…

12 questions · Seed 42 · Age guidance, not curriculum certification

View the answer key

Worksheet StudioAnswer key · Ages 16-18

Binomial theorem coefficients

Answers and reasoning. Review before teaching.

1. Find the coefficient of x^1 in the binomial expansion of (x + 3)^4.
108
By the binomial theorem, the x^1 term has coefficient C(4, 3) × 3^3 = 4 × 27 = 108.
2. Find the coefficient of x^3 in the binomial expansion of (x + 1)^6.
20
By the binomial theorem, the x^3 term has coefficient C(6, 3) × 1^3 = 20 × 1 = 20.
3. Find the coefficient of x^2 in the binomial expansion of (x + 4)^3.
12
By the binomial theorem, the x^2 term has coefficient C(3, 1) × 4^1 = 3 × 4 = 12.
4. Find the coefficient of x^3 in the binomial expansion of (x + 3)^6.
540
By the binomial theorem, the x^3 term has coefficient C(6, 3) × 3^3 = 20 × 27 = 540.
5. Find the coefficient of x^5 in the binomial expansion of (x + 2)^6.
12
By the binomial theorem, the x^5 term has coefficient C(6, 1) × 2^1 = 6 × 2 = 12.
6. Find the coefficient of x^1 in the binomial expansion of (x + 4)^6.
6144
By the binomial theorem, the x^1 term has coefficient C(6, 5) × 4^5 = 6 × 1024 = 6144.
7. Find the coefficient of x^1 in the binomial expansion of (x + 4)^4.
256
By the binomial theorem, the x^1 term has coefficient C(4, 3) × 4^3 = 4 × 64 = 256.
8. Find the coefficient of x^4 in the binomial expansion of (x + 7)^5.
35
By the binomial theorem, the x^4 term has coefficient C(5, 1) × 7^1 = 5 × 7 = 35.
9. Find the coefficient of x^1 in the binomial expansion of (x + 6)^4.
864
By the binomial theorem, the x^1 term has coefficient C(4, 3) × 6^3 = 4 × 216 = 864.
10. Find the coefficient of x^4 in the binomial expansion of (x + 1)^6.
15
By the binomial theorem, the x^4 term has coefficient C(6, 2) × 1^2 = 15 × 1 = 15.
11. Find the coefficient of x^1 in the binomial expansion of (x + 5)^4.
500
By the binomial theorem, the x^1 term has coefficient C(4, 3) × 5^3 = 4 × 125 = 500.
12. Find the coefficient of x^3 in the binomial expansion of (x + 5)^4.
20
By the binomial theorem, the x^3 term has coefficient C(4, 1) × 5^1 = 4 × 5 = 20.

12 questions · Seed 42 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 16-18

Binomial theorem coefficients

Take your time. Show your thinking.

1. Find the coefficient of x^1 in the binomial expansion of (x + 3)^4.
1 · 3 · 4 · Binomial theorem coefficie…
2. Find the coefficient of x^3 in the binomial expansion of (x + 1)^6.
3 · 1 · 6 · Binomial theorem coefficie…
3. Find the coefficient of x^2 in the binomial expansion of (x + 4)^3.
2 · 4 · 3 · Binomial theorem coefficie…
4. Find the coefficient of x^3 in the binomial expansion of (x + 3)^6.
3 · 3 · 6 · Binomial theorem coefficie…
5. Find the coefficient of x^5 in the binomial expansion of (x + 2)^6.
5 · 2 · 6 · Binomial theorem coefficie…
6. Find the coefficient of x^1 in the binomial expansion of (x + 4)^6.
1 · 4 · 6 · Binomial theorem coefficie…
7. Find the coefficient of x^1 in the binomial expansion of (x + 4)^4.
1 · 4 · 4 · Binomial theorem coefficie…
8. Find the coefficient of x^4 in the binomial expansion of (x + 7)^5.
4 · 7 · 5 · Binomial theorem coefficie…
9. Find the coefficient of x^1 in the binomial expansion of (x + 6)^4.
1 · 6 · 4 · Binomial theorem coefficie…
10. Find the coefficient of x^4 in the binomial expansion of (x + 1)^6.
4 · 1 · 6 · Binomial theorem coefficie…
11. Find the coefficient of x^1 in the binomial expansion of (x + 5)^4.
1 · 5 · 4 · Binomial theorem coefficie…
12. Find the coefficient of x^3 in the binomial expansion of (x + 5)^4.
3 · 5 · 4 · Binomial theorem coefficie…

12 questions · Seed 42 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 16-18

Binomial theorem coefficients

Answers and reasoning. Review before teaching.

1. Find the coefficient of x^1 in the binomial expansion of (x + 3)^4.
108
By the binomial theorem, the x^1 term has coefficient C(4, 3) × 3^3 = 4 × 27 = 108.
2. Find the coefficient of x^3 in the binomial expansion of (x + 1)^6.
20
By the binomial theorem, the x^3 term has coefficient C(6, 3) × 1^3 = 20 × 1 = 20.
3. Find the coefficient of x^2 in the binomial expansion of (x + 4)^3.
12
By the binomial theorem, the x^2 term has coefficient C(3, 1) × 4^1 = 3 × 4 = 12.
4. Find the coefficient of x^3 in the binomial expansion of (x + 3)^6.
540
By the binomial theorem, the x^3 term has coefficient C(6, 3) × 3^3 = 20 × 27 = 540.
5. Find the coefficient of x^5 in the binomial expansion of (x + 2)^6.
12
By the binomial theorem, the x^5 term has coefficient C(6, 1) × 2^1 = 6 × 2 = 12.
6. Find the coefficient of x^1 in the binomial expansion of (x + 4)^6.
6144
By the binomial theorem, the x^1 term has coefficient C(6, 5) × 4^5 = 6 × 1024 = 6144.
7. Find the coefficient of x^1 in the binomial expansion of (x + 4)^4.
256
By the binomial theorem, the x^1 term has coefficient C(4, 3) × 4^3 = 4 × 64 = 256.
8. Find the coefficient of x^4 in the binomial expansion of (x + 7)^5.
35
By the binomial theorem, the x^4 term has coefficient C(5, 1) × 7^1 = 5 × 7 = 35.
9. Find the coefficient of x^1 in the binomial expansion of (x + 6)^4.
864
By the binomial theorem, the x^1 term has coefficient C(4, 3) × 6^3 = 4 × 216 = 864.
10. Find the coefficient of x^4 in the binomial expansion of (x + 1)^6.
15
By the binomial theorem, the x^4 term has coefficient C(6, 2) × 1^2 = 15 × 1 = 15.
11. Find the coefficient of x^1 in the binomial expansion of (x + 5)^4.
500
By the binomial theorem, the x^1 term has coefficient C(4, 3) × 5^3 = 4 × 125 = 500.
12. Find the coefficient of x^3 in the binomial expansion of (x + 5)^4.
20
By the binomial theorem, the x^3 term has coefficient C(4, 1) × 5^1 = 4 × 5 = 20.

12 questions · Seed 42 · Age guidance, not curriculum certification