Which of the following terms appear in the expansion of \((x+y)^8\)?

Which of the following terms appear in the expansion of \((x+y)^8\)?

December 13, 2024

Question: Which of the following terms appear in the expansion of (x+y)8(x+y)^8? The letter aa in each term represents a real constant.

A. ay4ay^4
B. ax4y4ax^4y^4
C. ax4ax^4
D. ax3y5ax^3y^5

Answer:

B. ax4y4ax^4y^4 and D. ax3y5ax^3y^5

Explanation:

Step1: Understanding the Binomial Expansion of (x+y)8(x + y)^8

The binomial expansion of (x+y)n(x + y)^n follows the Binomial Theorem, which states:

(x+y)n=k=0n(nk)xnkyk(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k}

Where:

  • (nk)\binom{n}{k} is the binomial coefficient, calculated as n!k!(nk)!\frac{n!}{k!(n-k)!}.
  • kk ranges from 0 to nn.

For (x+y)8(x + y)^8, the expansion will consist of terms where the exponents of xx and yy add up to 8.

Step2: Analyzing Each Option

  • Option A: ay4ay^4
    Analysis:
    The term ay4ay^4 implies that the exponent of yy is 4 and the exponent of xx is 0 (since xx is not present). Therefore, the total degree of the term is 0+4=40 + 4 = 4.
    Conclusion:
    Incorrect. In the expansion of (x+y)8(x + y)^8, all terms must have exponents of xx and yy that sum to 8. This term only sums to 4.

  • Option B: ax4y4ax^4y^4
    Analysis:
    Here, the exponents of xx and yy are both 4, summing to 4+4=84 + 4 = 8.
    Conclusion:
    Correct. This term fits the requirement of the binomial expansion of (x+y)8(x + y)^8.

  • Option C: ax4ax^4
    Analysis:
    The term ax4ax^4 implies that the exponent of xx is 4 and the exponent of yy is 0. The total degree is 4+0=44 + 0 = 4.
    Conclusion:
    Incorrect. The total degree does not equal 8.

  • Option D: ax3y5ax^3y^5
    Analysis:
    In this term, the exponent of xx is 3 and the exponent of yy is 5, summing to 3+5=83 + 5 = 8.
    Conclusion:
    Correct. This term is a valid part of the binomial expansion of (x+y)8(x + y)^8.

Extended Knowledge:

Binomial Expansion and the Binomial Theorem

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:

(x+y)n=k=0n(nk)xnkyk(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k}

Where:

  • (nk)\binom{n}{k} is the binomial coefficient, representing the number of ways to choose kk elements from a set of nn elements without regard to order.
  • xx and yy are variables.
  • nn is a non-negative integer representing the power to which the binomial is raised.

Properties of Terms in the Expansion

  1. Degree of Terms:
    Each term in the expansion has a total degree equal to the power nn. For (x+y)8(x + y)^8, every term must have exponents that add up to 8.

  2. Binomial Coefficients:
    The coefficients of the terms are given by the binomial coefficients (8k)\binom{8}{k}, where kk ranges from 0 to 8. These coefficients determine the number of ways to arrange the terms in the expansion.

  3. Symmetry:
    The expansion is symmetrical. The term with xkynkx^k y^{n-k} has the same coefficient as the term with xnkykx^{n-k} y^k.

Practical Application Example

Example:
Expand (x+y)3(x + y)^3 using the Binomial Theorem.

(x+y)3=(30)x3y0+(31)x2y1+(32)x1y2+(33)x0y3(x + y)^3 = \binom{3}{0}x^3y^0 + \binom{3}{1}x^2y^1 + \binom{3}{2}x^1y^2 + \binom{3}{3}x^0y^3 =1x3+3x2y+3xy2+1y3= 1 \cdot x^3 + 3 \cdot x^2y + 3 \cdot xy^2 + 1 \cdot y^3 =x3+3x2y+3xy2+y3= x^3 + 3x^2y + 3xy^2 + y^3

This demonstrates how each term's exponents sum to 3, consistent with the power of the binomial.

Identifying Valid Terms in Higher Powers

When dealing with higher powers, such as (x+y)8(x + y)^8, 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:

  1. Check the Total Degree:
    Sum the exponents of xx and yy in the term. It must equal 8.

  2. Match with Binomial Coefficients:
    Ensure the coefficient aligns with (8k)\binom{8}{k}, where kk is the exponent of yy.

  3. Simplify Radicals (if present):
    In some expansions, terms may include radicals or constants. Ensure they are correctly simplified and incorporated.

Common Mistakes to Avoid

  1. Incorrect Total Degree:
    Including terms where the exponents do not sum to the binomial's power, as seen in Options A and C.

  2. Miscalculating Binomial Coefficients:
    Assigning incorrect coefficients to terms, which can lead to errors in the expansion.

  3. Ignoring Symmetry:
    Overlooking the symmetrical nature of the expansion can result in incomplete or incorrect terms.

Advanced Considerations

  • Pascal's Triangle:
    A useful tool for quickly determining binomial coefficients for small powers. Each row corresponds to the coefficients of (x+y)n(x + y)^n.

  • 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.

Conclusion

In the expansion of (x+y)8(x + y)^8, each term must have exponents of xx and yy that add up to 8. Therefore, only the terms ax4y4ax^4y^4 and ax3y5ax^3y^5 (Options B and D) satisfy this condition and appear in the expansion. Recognizing the structure of binomial expansions ensures accurate identification of valid terms.