A.
B.
C.
D.
B. and D.
The binomial expansion of follows the Binomial Theorem, which states:
Where:
For , the expansion will consist of terms where the exponents of and add up to 8.
Option A:
Analysis:
The term implies that the exponent of is 4 and the exponent of is 0 (since is not present). Therefore, the total degree of the term is .
Conclusion:
Incorrect. In the expansion of , all terms must have exponents of and that sum to 8. This term only sums to 4.
Option B:
Analysis:
Here, the exponents of and are both 4, summing to .
Conclusion:
Correct. This term fits the requirement of the binomial expansion of .
Option C:
Analysis:
The term implies that the exponent of is 4 and the exponent of is 0. The total degree is .
Conclusion:
Incorrect. The total degree does not equal 8.
Option D:
Analysis:
In this term, the exponent of is 3 and the exponent of is 5, summing to .
Conclusion:
Correct. This term is a valid part of the binomial expansion of .
The Binomial Theorem is a powerful tool in algebra that provides a formula for expanding expressions that are raised to any positive integer power. The theorem is expressed as:
Where:
Degree of Terms:
Each term in the expansion has a total degree equal to the power . For , every term must have exponents that add up to 8.
Binomial Coefficients:
The coefficients of the terms are given by the binomial coefficients , where ranges from 0 to 8. These coefficients determine the number of ways to arrange the terms in the expansion.
Symmetry:
The expansion is symmetrical. The term with has the same coefficient as the term with .
Example:
Expand using the Binomial Theorem.
This demonstrates how each term's exponents sum to 3, consistent with the power of the binomial.
When dealing with higher powers, such as , it's essential to ensure that the exponents in each term sum to 8. Terms that do not meet this criterion are not part of the expansion.
Key Steps to Identify Valid Terms:
Check the Total Degree:
Sum the exponents of and in the term. It must equal 8.
Match with Binomial Coefficients:
Ensure the coefficient aligns with , where is the exponent of .
Simplify Radicals (if present):
In some expansions, terms may include radicals or constants. Ensure they are correctly simplified and incorporated.
Incorrect Total Degree:
Including terms where the exponents do not sum to the binomial's power, as seen in Options A and C.
Miscalculating Binomial Coefficients:
Assigning incorrect coefficients to terms, which can lead to errors in the expansion.
Ignoring Symmetry:
Overlooking the symmetrical nature of the expansion can result in incomplete or incorrect terms.
Pascal's Triangle:
A useful tool for quickly determining binomial coefficients for small powers. Each row corresponds to the coefficients of .
Applications in Probability:
Binomial expansions are foundational in calculating probabilities in binomial distributions, where each term represents the probability of a specific number of successes in a series of independent trials.
Algebraic Manipulations:
Understanding binomial expansions aids in simplifying complex algebraic expressions and solving polynomial equations.
In the expansion of , each term must have exponents of and that add up to 8. Therefore, only the terms and (Options B and D) satisfy this condition and appear in the expansion. Recognizing the structure of binomial expansions ensures accurate identification of valid terms.