The constant term in the expansion of (2x + 1/x⁷ + 3x²)⁵ is ______.

Q: Wait, why can't we just use the Binomial Theorem here?

The Binomial Theorem is specifically designed for expressions with two terms, like $(a+b)^n$. Since our expression $(2x + \frac{1}{x^7} + 3x^2)^5$ has three terms, we must use the Multinomial Theorem, which is the natural generalization of the binomial expansion.

Q: Why does the sum of powers $n_1 + n_2 + n_3$ have to equal $5$?

In the multinomial expansion of $(a+b+c)^n$, every term is of the form $\frac{n!}{n_1! n_2! n_3!} a^{n_1} b^{n_2} c^{n_3}$. The sum of the exponents $n_1 + n_2 + n_3$ must always equal the total power $n$ of the expression. Here, $n=5$, so $n_1 + n_2 + n_3 = 5$ is a mandatory constraint.

Q: How do we know which integer values to pick for $n_2$?

After setting up the system of equations, we found a relationship $n_3 = 8n_2 - 5$. Since $n_1, n_2,$ and $n_3$ must be non-negative integers and their sum is $5$, we test values for $n_2$. If $n_2=0$, $n_3=-5$ (impossible). If $n_2=1$, $n_3=3$ (valid). If $n_2=2$, $n_3=11$ (impossible, as $n_3$ cannot exceed $5$). Thus, $n_2=1$ is the only logical choice.

Question

The constant term in the expansion of $(2x + \frac{1}{x^7} + 3x^2)^5$ is ______.

Sigma Percentile · Accepted Answer

### The Beauty of the Multinomial Theorem

Welcome, aspiring mathematicians! Today, we are going to peel back the layers of a seemingly intimidating algebraic expression. We are tasked with finding the constant term in the expansion of $(2x + x^{-7} + 3x^2)^5$.

When you see an expression with three terms raised to a power, your first instinct might be to panic, but I want you to take a deep breath. This is not a monster; it is a beautiful puzzle waiting to be solved by the **Multinomial Theorem**.

### Phase 1: Deconstructing the Trinomial

First, let us define what a 'constant term' actually is. In the world of polynomials, a constant term is simply the term that is independent of $x$. Mathematically, this means we are hunting for the coefficient of $x^0$.

To find this, we use the Multinomial Theorem. For an expression $(a + b + c)^n$, the general term is given by:

$$T = \frac{n!}{n_1! n_2! n_3!} a^{n_1} b^{n_2} c^{n_3}$$

In this formula, we have the strict constraint that $n_1 + n_2 + n_3 = n$. In our specific case, $a = 2x$, $b = x^{-7}$, $c = 3x^2$, and $n = 5$.

Substituting these values, our general term $T$ becomes:

$$T = \frac{5!}{n_1! n_2! n_3!} (2x)^{n_1} (x^{-7})^{n_2} (3x^2)^{n_3}$$

### Phase 2: The Power Hunt

Now, let us isolate the constants from the variables. We pull out the coefficients $2^{n_1}$ and $3^{n_3}$. The variable part is $x^{n_1} \cdot (x^{-7})^{n_2} \cdot (x^2)^{n_3}$.

Using the laws of exponents, we combine these into $x^{n_1 - 7n_2 + 2n_3}$. For this to be a constant term, the exponent of $x$ must be zero.

This gives us our first crucial equation:

$$n_1 - 7n_2 + 2n_3 = 0$$

Combined with our constraint $n_1 + n_2 + n_3 = 5$, we have a system of two equations with three variables.

### Phase 3: Solving the System

From the first equation, we can write $n_1 = 7n_2 - 2n_3$. Substituting this into the second equation, we get:

$$(7n_2 - 2n_3) + n_2 + n_3 = 5$$

This simplifies to $8n_2 - n_3 = 5$, or $n_3 = 8n_2 - 5$. This is where the magic happens. We need $n_1, n_2,$ and $n_3$ to be non-negative integers.

If we test $n_2 = 0$, we get $n_3 = -5$, which is impossible. If we test $n_2 = 1$, we get $n_3 = 3$, which is a valid, positive integer. If we test $n_2 = 2$, we get $n_3 = 11$, which is impossible because the sum of powers cannot exceed $5$.

Thus, we have found our unique solution: $n_2 = 1$ and $n_3 = 3$. Plugging these back into $n_1 + n_2 + n_3 = 5$, we find $n_1 = 1$.

### Phase 4: The Grand Finale

With our powers $(n_1, n_2, n_3) = (1, 1, 3)$ identified, we calculate the coefficient:

$$\frac{5!}{1! 1! 3!} 	imes 2^1 	imes 3^3$$

This simplifies to:

$$\frac{120}{6} 	imes 2 	imes 27 = 20 	imes 2 	imes 27 = 1080$$

And there you have it! The constant term is **1080**. You have successfully navigated the multinomial landscape; keep this logic in your toolkit, and no expansion will ever intimidate you again.

Animated Solution for Mathematics - Binomial Theorem: The constant term in the expansion of $(2 x + \frac{1}{x ^{7}} + 3 x^{2})^{5}$ is ______.

Enter Numerical Value:

Visualized Solution

Identify the Expression

The Multinomial Theorem

Defining our Variables

Separating Constants and Variables

The Constant Term Condition

Eliminating $n_{1}$

Finding the Relation

Testing Integer Constraints

Determining $n_{1}$

Calculating the Coefficient

Final Result

The Sigma Insight: Multinomial Theorem

The Beauty of the Multinomial Theorem

Phase 1

Phase 2

Phase 3

Phase 4

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Animated Solution for Mathematics - Binomial Theorem: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The constant term in the expansion of (2x+x71​+3x2)5 is ______.