Problem 52 - Precalculus with Limits

Question

In Exercises 45 - 52, find the specified \( n \)th term in the expansion of the binomial. \( \left(7x + 2y\right)^{15}, \quad n = 7 \)

Step-by-Step Solution

Answer

The seventh term in the expansion of the binomial \( (7x + 2y)^{15} \) is \( 2584787472100667392 x^9 y^6 \).

Step 1: Understanding the Binomial Theorem and the expression for the nth term

The general term in the expansion of \( (a+b)^n \) using the binomial theorem is given by: \[ T_{r+1} = ^nC_r \cdot a^{n-r} \cdot b^r \] where \( T_{r+1} \) is the (r+1)th term, \( ^nC_r \) is the binomial coefficient, and r starts from 0. For our problem, a = 7x, b = 2y, and n = 15, and we're asked to find the 7th term, so r = 7 - 1 = 6.

Step 2: Substituting the values into the general term formula

Substitute the values into the formula for \( T_{r+1} \). Therefore, the 7th term of the binomial expansion is: \[ T_{7} = ^{15}C_{6} \cdot (7x)^{15-6} \cdot (2y)^6 \]

Step 3: Simplifying the expression

The binomial coefficient ^15C6 = \(\frac{15!}{6!(15-6)!} = 5005\). We then simplify the power expressions: (7x)^9 and (2y)^6. Thus, the expression can be written as \[ T_{7} = 5005 \cdot (7x)^9 \cdot (2y)^6 \] or \[ T_{7} = 5005 \cdot 7^9 \cdot x^9 \cdot 2^6 \cdot y^6 \]

Step 4: Evaluating the expression

We now calculate the product to obtain the 7th term of the expansion, which gives \[ T_{7} = 2584787472100667392 x^9 y^6 \]

Key Concepts

Binomial Theorem

The binomial theorem is a powerful method for expanding expressions that are raised to a power. It allows us to express a binomial, which is a sum of two terms, raised to a large exponent as a sum of terms involving coefficients, powers of the first term, and powers of the second term. This makes computing large powers much simpler.

For example, when we have an expression like \((a + b)^n\), the binomial theorem provides us with a formula to find each term in the expansion. The general form of each term is given by:

• \(T_{r+1} = \binom{n}{r} \cdot a^{n-r} \cdot b^r \)

In this formula, \(\binom{n}{r}\) represents a binomial coefficient, and it defines the weighting factor for each of the terms.

By using the binomial theorem, complex expressions like \((7x + 2y)^{15}\) become straightforward to handle, as demonstrated in the exercise.

Binomial Coefficient

The binomial coefficient is an essential component of the binomial theorem. It is a number that appears in the polynomial expansion of a binomial. Denoted by \(\binom{n}{r}\), this coefficient gives us the number of ways to choose \(r\) elements from a set of \(n\) elements without regard to the order of selection.

Mathematically, the binomial coefficient is calculated using the formula:

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

Where \(!\) denotes a factorial, meaning the product of all positive integers up to that number.

In the context of our exercise, the binomial coefficient \(\binom{15}{6}\) calculates to 5005. This means that there are 5005 ways to select 6 elements from a set of 15. It shows just how many times that specific term's contribution is magnified in the final expansion of the binomial expression.

Polynomial Terms

Polynomial terms are the building blocks of polynomial expressions. Each term is a product of constants, variables, and exponents. In a binomial expansion, each of these terms results from applying the binomial theorem repeatedly.

For the specific problem \((7x + 2y)^{15}\), each term has a structure of a polynomial due to the nature of the binomial expansion. The powers of each variable decrease or increase according to the positions defined by the formula.

• The term \( (7x)^{15-6} \cdot (2y)^6 \) shows how the power of 7x decreases while 2y increases.

Understanding polynomial terms helps in simplifying and calculating large polynomial expressions efficiently. It enables you to break down the larger powers into manageable and computable pieces, making complicated algebraic problems much more approachable.