Binomial Expansion Examples and Practice Questions PDF
(5) Find the full expansion of:
@(2+x)^4
Step 1: Write the bracket in the form (a+b)^n.
(2 + x)^2 = (a + b)^2 where a = 2 and b = x.
Step 2: Write the full expansion of (a+b)^n leaving the "Cr gaps.
(a+b)^4 = 4C0a^4 + 4C1a^3b + 4C2a^2b^2 + 4C3ab^3 + 4C4b^4.
Step 3: Use the "Cr" button to calculate the terms that will be used as coefficients for a and b.
- For a:
- Coefficient for a^2 = 1
- Coefficient for a^3 = 4
- Coefficient for a^2b^2 = 6
- Coefficient for ab^3 = 4
- Coefficient for b^4 = 1
Step 4: Write in full the binomial expansion of (a+b)^4
(a+b)^4 = 1a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + 1b^4
- The full expansion is: 1 + 4x + 6x^2 + 4x^3 + x^4.
Step 5: Sub in a = 2 and b = x into the above expansion
(2 + x)^2 = 1(2)^4 + 4(2)^3(x) + 6(2)^2(x)^2 + 4(2)(x)^3 + 1(x)^4
= 16 + 32x + 24x^2 + 8x^3 + x^4
(Q4) The term in x^2 for the expansion of (1+x)^n is 231x^2.
(a) What is the value of n?
Step 1: Write (1+x)^n in the form (a+b)^n
(1+x)^n = (a + b)^n where a = 1, b = x, and n = n.
Step 2: Expand (a+b)^n until you reach b^2 (leave the "Cr" spaces blank)
(a+b)^n = a^0 + a^1b + a^2b^2
Step 3: Fill in the "Cr" gaps
- 1st term = 1
- 2nd term = n
- 3rd term = 231
Step 4: Use the 3rd (b^2) term to calculate the value of n
231x^2 = "C2 x^2
231 = "C2
As "C1 = n!,
the b^2 term is the term that will have x^2, so b^2 = x^2 and a = 1.
Step 5: Solve for n
231 = n!(2)! (n-2)!
462 = n!(n-2)!
(n-2)! = 1
462 = n
The value of n is 22.
(2+2x)^4
Step 1: Write the bracket in the form (a+b)^n
(2+2x)^4
Step 2: Write the expansion of (a+b)^n leaving the "Cr" gaps blank
(a+b)^4 = 4C0a^4 + 4C1a^3b + 4C2a^2b^2 + 4C3ab^3 + 4C4b^4
Step 3: Use the "Cr" button to calculate the terms that will be used as coefficients for a and b
- Coefficient for a^4 = 1
- Coefficient for a^3b = 4
- Coefficient for a^2b^2 = 6
- Coefficient for ab^3 = 4
- Coefficient for b^4 = 1
Step 4: Write in full the binomial expansion of (a+b)^4
(a+b)^4 = 1a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + 1b^4
Step 5: Sub in a=2 and b=2x into the above expansion
(2+2x)^4 = 1(2)^4 + 4(2)^3(2x) + 6(2)^2(2x)^2 + 4(2)(2x)^3 + 1(2x)^4
= 16 + 64x + 96x^2 + 64x^3 + 16x^4
Binomial Theorem and Expansion
Finding Full Expansion of Binomial Expressions
Example:
- (2x-3)^9= 512x^9 − 6912 x^7 + 41472x^5 − 147456x^3 + 196608x
Binomial Expansion of (a+b)
The "Binomial" means terms of two, for example:
- (a+b)
- (x+y)
Watch Video Tutorials on Expanding Binomials
Watch the following videos to see how we create the formula used to expand binomial:
By practicing binomial expansion examples and working through binomial expansion questions and answers, you can improve your understanding of the binomial theorem and its applications. Utilize the binomial expansion examples and practice questions PDF to further enhance your skills.
Conclusion
In conclusion, understanding binomial expansion is essential in various fields such as mathematics, statistics, and engineering. The ability to find the coefficient in binomial expansion, as well as the full expansion of binomial expressions, is a valuable skill that can be applied in real-world problem-solving and analysis. With practice and dedication, mastering the concepts of binomial expansion will open up a world of possibilities for analytical problem-solving and mathematical applications.
