Question

Find the coefficient of \(\mathrm{x}^{4}\) in the expansion of \(\left(2 \mathrm{x}^{2}+\frac{3}{\mathrm{x}^{3}}\right)^{7}\). (1) \({ }^{7} \mathrm{C}_{2} 2^{5} 3^{3}\) (2) \({ }^{7} \mathrm{C}_{2} 2^{5} 3^{2}\) (3) \({ }^{7} \mathrm{C}_{2} 3^{5} 2^{2}\) (4) \({ }^{7} \mathrm{C}_{3} 2^{5} 3^{2}\)

Short answer

Answer: 6048.

Step-by-step solution

Know the formula

Recall the Binomial theorem formula for the expansion of (a + b)^n: \((a + b)^n = \sum_{r=0}^n {n \choose r} \cdot a^{n-r} \cdot b^r\), where \({n \choose r}\) denotes the binomial coefficient.

Identify the given terms

In our case, n = 7 and we have the terms: a = 2x^2 and b = 3/x^3.

Apply the formula and find x^4 term

We need to find the term with x^4 in the expansion of (2x^2 + 3/x^3)^7. We do this by finding the values of r for which the power of x in the term is 4. Using the Binomial theorem formula, the term Tr+1 with x^4 will be: \({7 \choose r} \cdot (2x^2)^{7-r} \cdot (3/x^3)^r\). Let's find the power of x in this term: \(2(7 - r) - 3r = 4 \Rightarrow r = 2\). Now that we found the required r-value, let's find the term.

Compute the term

Substitute the value of r in the term: \({7 \choose 2} \cdot (2x^2)^{7-2} \cdot (3/x^3)^2\).

Calculate the coefficient of x^4

Now, we need to find the coefficient by solving the expression: \({7 \choose 2} \cdot 2^{5} \cdot 3^{2}\). We know that \({7 \choose 2} = \frac{7!}{5!2!} = \frac{7 \cdot 6}{2} = 21\). So, the coefficient of x^4 is: \(21 \cdot 2^5 \cdot 3^2 = 21 \cdot 32 \cdot 9 = 6048\). Thus, the correct answer is \({ }^{7} \mathrm{C}_{2} 2^{5} 3^{2}\) (Option 2).

Key concepts

Binomial Expansion

The binomial expansion is a way of expressing a binomial expression raised to a power as a sum of terms. This is accomplished through the application of the binomial theorem. For instance, consider an expression of the form \( (a + b)^n \). To expand this expression, we employ the binomial theorem, which states that the expansion will lead to a series of terms that include coefficients known as binomial coefficients. These coefficients can be found using the formula \( {n \choose r} \), where 'n' is the power, and 'r' is the term number, starting from 0.

Each term in the expansion is a combination of powers of 'a' and 'b', with the powers of 'a' decreasing and the powers of 'b' increasing as we move across the terms. Mathematically, the binomial expansion is written as: \[ (a + b)^n = \sum_{r=0}^n {n \choose r} \cdot a^{n-r} \cdot b^r \].

By fully understanding binomial expansion, students can efficiently work with polynomial expressions and solve complex algebraic problems involving exponents.

Polynomial Coefficient

In the context of polynomials and binomial expansions, the coefficient is a crucial concept that refers to a numerical factor that multiplies a given term in a polynomial. It's essential to differentiate between terms and their coefficients. For example, in the term \( 5x^3 \), '5' is the coefficient, while \( x^3 \) signifies the term's variable part. In a binomial expansion, focusing on the polynomial coefficients helps us find specific terms of interest.

When we calculate the coefficient of a particular power of 'x' in a binomial expression, we're essentially determining the numerical multiplier of that term once the expression has been expanded. It requires combining the binomial coefficient calculated from the binomial theorem, and any constants or additional powers that may be part of the term. This process involves algebraic manipulation and understanding the properties of exponents and factorials. Coefficient calculation plays a vital role in solving equations, simplifying expressions, and in various applications across mathematics and science.

Mathematical Induction

Mathematical induction is a powerful and fundamental proof technique used in mathematics, particularly to establish the validity of statements that depend on a natural number 'n'. The process follows a two-step approach where the first step, called the base case, verifies that the statement holds true for the first natural number in the series, typically \( n = 1 \).

In the second step, known as the induction step, we assume that the statement holds for a certain natural number 'k', and then prove that it must also be true for the next number, which is 'k+1'. This method utilizes the principle that if both steps succeed, then the statement can be concluded to be true for all natural numbers. Mathematical induction is particularly useful when dealing with sequential patterns or properties related to series and sequences. It might not directly relate to finding polynomial coefficients or binomial expansion, but understanding this method enriches one's problem-solving skills and logical reasoning in mathematics.