Question

One term in the expansion of \((2 x-m)^{7}\) is \(-15120 x^{4} y^{3} .\) Determine \(m\).

Short answer

The value of \( m \) is 3.

Step-by-step solution

Identify the General Term

The general term in the expansion of \( (a + b)^n \) is given by \[ T_k = \binom{n}{k} a^{n-k} b^k \] where \( n \) is the exponent, \( a \) is the first term, and \( b \) is the second term. For \( (2x - m)^7, a = 2x \) and \( b = -m \), and \( n = 7 \).

Set Up the Term That Corresponds to \( -15120 x^4 y^3 \)

We want to find the specific term where \( x \) is raised to the 4th power. Use the general term expression: \[ T_k = \binom{7}{k} (2x)^{7-k} (-m)^k \] Given \( x^4 \), it implies that \( 7 - k = 4 \), thus, \ k = 3 \.

Substitute \( k = 3 \) and Calculate the Term

Substitute \( k = 3 \) into the general term formula: \[ T_3 = \binom{7}{3} (2x)^{7-3} (-m)^3 = \binom{7}{3} (2x)^4 (-m)^3 = \binom{7}{3} (16x^4) (-m)^3 \]

Evaluate Binomial Coefficient and Simplify

Calculate the binomial coefficient: \[ \binom{7}{3} = \frac{7!}{3! (7-3)!} = 35 \] Substitute and simplify: \[ T_3 = 35 \times 16 x^4 \times (-m)^3 = 35 \times 16 x^4 \times (-m)^3 = 560 x^4 (-m)^3 = -560m^3 x^4 \]

Equate and Solve for \( m \)

We know \( -560 m^3 x^4 \) should equal \( -15120 x^4 y^3 \). Extract the coefficients and solve for \( m \): \[ -560 m^3 = -15120 \] Divide both sides by \( -560 \): \[ m^3 = \frac{15120}{560} = 27 \] Take the cube root of both sides: \[ m = \sqrt[3]{27} = 3 \]

Key concepts

General Term

The general term in the binomial expansion is crucial in finding particular terms of an expanded polynomial. In any binomial expansion of the form \( (a + b)^n \), the general term is expressed as \( T_k = \binom{n}{k} a^{n-k} b^k \). Here:

• \( n \) is the exponent
• \( a \) is the first term in the binomial
• \( b \) is the second term of the binomial
• \( k \) is the specific term's index you want to find. The coefficients generated by \( \binom{n}{k} \) are crucial, since they denote the number of ways an event can occur.

Understanding this can unlock any specific term you want inside the binomial expansion.

Binomial Coefficient

The binomial coefficient is a key concept when dealing with binomial expansions. It's denoted by \( \binom{n}{k} \) and also called 'n choose k'. It calculates the number of ways to choose k elements from a set of n elements without regard to order. The binomial coefficient in our formula is given by: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where \( ! \) denotes factorial, a product of an integer and all the integers below it. Simplifying the binomial coefficient is often the first step in solving binomial expansion problems. This work is often simplified with combinatorial rules to make calculations feasible, especially with larger values of n and k.

Polynomial Exponents

In binomial expansions, understanding polynomial exponents is essential. The exponents in the generalized term \( T_k = \binom{n}{k} a^{n-k} b^k \) suggest that as the index k changes, powers of a and b change dynamically. For example:

• When k = 0, the term becomes \( a^n \)
• When k = 1, the term includes \( a^{n-1}b \)
• When k = k, the term includes both a and b raised to their respective powers

If you carefully track how the exponents shift, you can pinpoint exactly which term in your binomial expansion corresponds to a certain polynomial structure, like we did in the given problem.

Solving Equations

The last step often involves solving simple algebraic equations. Once we've set up our generalized term and identified the coefficients, we equate them with actual known values. This equation must be solved to find unknown quantities. For the equation in the problem:

• Extract the known coefficients and equate them to those in your polynomial
• Isolate the variable, using basic algebraic operations like division or root extraction

For instance, solving for m:
Given \( -560 m^3 = -15120 \), we divided both sides by \( -560 \) and found the cube root of the result. This process usually simplifies to linear, quadratic, or cubic equations, depending on your polynomial.