Problem 38 - College Algebra

Question

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ \left(y^{3}-1\right)^{21} $$

Step-by-Step Solution

Answer

The first three terms of the binomial expansion are \( y^{63} \), \( -21*y^{60} \), \( 210*y^{57} \)

Step 1: First Term

The first term of the expansion is calculated by substituting \( k = 0 \) into the binomial theorem formula which gives us: \( {21 choose 0}*(y^3)^{21}*(-1)^0 = y^{63} \)

Step 2: Second Term

The second term of the expansion is calculated by substituting \( k = 1 \) into the binomial theorem formula which gives us: \( {21 choose 1}*(y^3)^{20}*(-1)^1 = -21*y^{60} \)

Step 3: Third Term

The third term of the expansion is calculated by substituting \( k = 2 \) into the binomial theorem formula which gives us: \( {21 choose 2}*(y^3)^{19}*(-1)^2 = 210*y^{57} \)

Key Concepts

Binomial Theorem

The binomial theorem is a powerful tool in algebra used to expand expressions of the form \((a + b)^n\). It provides a way to express these powers as a sum of terms involving products of binomial coefficients and powers of the two terms involved. In our exercise, we're using the binomial theorem to expand \((y^3 - 1)^{21}\).

Each term in a binomial expansion has the form \(\binom{n}{k} a^{n-k} b^k\), where:

• \(n\) is the power to which the binomial is raised.
• \(k\) is the current term number starting from zero.
• \(\binom{n}{k}\) is the combinatorial coefficient ("n choose k").
• \(a\) and \(b\) are the two parts of the binomial.

For our binomial \((y^3 - 1)\), \(a = y^3\) and \(b = -1\). Plugging these into the theorem provides a structured way to calculate each term of the expansion.

Combinatorial Coefficients

Combinatorial coefficients, often referred to as "binomial coefficients," are the numbers that appear in the binomial theorem's expansion. They are represented by \(\binom{n}{k}\) and calculated using the formula:

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

These coefficients tell us how many different ways we can choose \(k\) elements out of a set of \(n\) elements. In the context of the binomial expansion, they help determine the weighting of each term in the expanded expression.

For example, in our exercise:

• \(\binom{21}{0} = 1\)
• \(\binom{21}{1} = 21\)
• \(\binom{21}{2} = 210\)

Each term gets multiplied by its respective binomial coefficient, significantly impacting the growth or reduction of each term's magnitude in the polynomial expansion.

Polynomial Expressions

A polynomial expression consists of variables and coefficients, expressed in terms of power sums. In our exercise, the binomial expansion results in generating a polynomial expression from the binomial \((y^3 - 1)^{21}\).

After applying the binomial theorem and calculating each of the first three terms:

• The first term: \(y^{63}\)
• The second term: \(-21y^{60}\)
• The third term: \(210y^{57}\)

These terms collectively form part of the polynomial expression resulting from the expansion. Polynomial expressions can be quite large, especially when dealing with high powers, which is why leveraging the binomial theorem is so beneficial. It helps in systematically breaking down these expressions into manageable components based on established arithmetic rules.