The sum of the real values of x for which the middle term in the binomial expansion of (x³/3 + 3/x)⁸

Q: Wait, why is the middle term $T_5$? Shouldn't it be $T_4$ because $8/2 = 4$?

That is a classic trap! Remember, the index of the term is $r+1$. For $n=8$, we have 9 terms. The middle term is the 5th term. If you pick $T_4$, you are slightly off-center. Always visualize the sequence: 1, 2, 3, 4, **5**, 6, 7, 8, 9. The 5th term is the true center.

Q: Why did we get $x^2 = 3$ instead of $x^8 = 81$?

We didn't lose the power! We had $x^8 = 81$. Since $81 = 3^4$, we have $x^8 = 3^4$. Taking the fourth root of both sides gives $x^2 = 3$. It is a simplification step, not a loss of information. It makes solving for $x$ much easier!

Q: Why do we have two values for $x$? Can't we just take the positive root?

The question asks for the sum of *all* real values. When you solve $x^2 = 3$, you must account for both the positive and negative roots: $x = \sqrt{3}$ and $x = -\sqrt{3}$. Ignoring the negative root would lead to an incorrect sum. Always check for symmetry in your roots!

Question

The sum of the real values of $x$ for which the middle term in the binomial expansion of $(\frac{x^3}{3} + \frac{3}{x})^8$ equals 5670 is :

Options:

(A) 6

(B) 8

(C) 0

(D) 4

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The expression provided is $(\frac{x^3}{3} + \frac{3}{x})^8$. We are tasked with finding the middle term and solving for $x$ given a specific value for that term.

In any binomial expansion of the form $(a+b)^n$, the total number of terms is $n+1$. With $n=8$, we have $8+1 = 9$ terms.

To find the middle term, we look for the position that balances the sequence. For 9 terms, the middle is the 5th term, $T_5$, calculated as:

$$T_{middle} = \frac{n}{2} + 1 = \frac{8}{2} + 1 = 5$$

### The General Term as a Key

We utilize the General Term formula for a binomial expansion:

$$T_{r+1} = \binom{n}{r} a^{n-r} b^r$$

Here, $n=8$, $a = \frac{x^3}{3}$, and $b = \frac{3}{x}$. To find $T_5$, we set $r=4$. Substituting these values into the formula yields:

$$T_5 = \binom{8}{4} \left( \frac{x^3}{3} ight)^{8-4} \left( \frac{3}{x} ight)^4$$

$$T_5 = \binom{8}{4} \left( \frac{x^3}{3} ight)^4 \left( \frac{3}{x} ight)^4$$

### The Algebraic Dance

First, we calculate the binomial coefficient $\binom{8}{4}$:

$$\binom{8}{4} = \frac{8 	imes 7 	imes 6 	imes 5}{4 	imes 3 	imes 2 	imes 1} = 70$$

Next, we simplify the variable components:

$$T_5 = 70 	imes \frac{x^{12}}{3^4} 	imes \frac{3^4}{x^4}$$

Notice that the $3^4$ terms cancel out perfectly. Applying the laws of exponents to the $x$ terms, we get:

$$T_5 = 70 	imes x^{12-4} = 70x^8$$

### The Final Reveal

We are given that the middle term equals $5670$. We set up the following equation:

$$70x^8 = 5670$$

Dividing both sides by $70$:

$$x^8 = \frac{5670}{70} = 81$$

Since $81 = 3^4$, we have $x^8 = 3^4$. Taking the fourth root of both sides gives $x^2 = 3$, which results in two real solutions:

$$x = \sqrt{3} \quad 	ext{and} \quad x = -\sqrt{3}$$

The sum of these real values is:

$$\sqrt{3} + (-\sqrt{3}) = \mathbf{0}$$

Animated Solution for Mathematics - Binomial Theorem: The sum of the real values of $x$ for which the middle term in the binomial expansion of $(\frac{x ^{3}}{3} + \frac{3}{x})^{8}$ equals 5670 is :

Select Answer:

Visualized Solution

Identify the Exponent $n$

Locate the Middle Term

General Term Formula

Substitute Values into $T_{5}$

Calculate the Coefficient $(4 8)$

Simplify Powers of $3$

Simplify Powers of $x$

Equate to Given Value

Solve for $x^{8}$

Find Real Values of $x$

Final Sum of Values

The Sigma Insight: General Term and Middle Term

Analyzing the Setup

The General Term as a Key

The Algebraic Dance

The Final Reveal

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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 sum of the real values of x for which the middle term in the binomial expansion of (3x3​+x3​)8 equals 5670 is :

Select Answer:

Visualized Solution

Identify the Exponent .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n

Locate the Middle Term

General Term Formula

Substitute Values into .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } T5​

Calculate the Coefficient .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (48​)

Simplify Powers of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 3

Simplify Powers of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x

Equate to Given Value

Solve for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x8

Find Real Values of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x