Problem 50 Use the binomial theorem to expa... [FREE SOLUTION]

Chapter 15: Problem 50

Use the binomial theorem to expand each expression. $$\left(t^{2}-\frac{1}{2} u\right)^{4}$$

Short Answer

Expert verified

The short version of the answer is: \[\left(t^2-\frac{1}{2}u\right)^4 = t^8 - 2t^6u + \frac{3}{2}t^4u^2 - \frac{1}{2}t^2u^3 + \frac{1}{16}u^4\]

Step by step solution

Identify the expression to expand

We have the following expression to expand: \[\left(t^{2}-\frac{1}{2}u\right)^4\]

Apply the binomial theorem

Using the binomial theorem, we can write the expression as: \[\left(t^{2} - \frac{1}{2}u\right)^4 = \sum_{k=0}^{4} \binom{4}{k} (t^2)^{4-k} \left(-\frac{1}{2}u\right)^k \]

Calculate the binomial coefficients and expand each term

Now, we will calculate the binomial coefficients and expand each term in the sum: \[\binom{4}{0} = 1\] \[\binom{4}{1} = 4\] \[\binom{4}{2} = 6\] \[\binom{4}{3} = 4\] \[\binom{4}{4} = 1\] Then, we find each term of the sum: \[1 \cdot (t^{2})^{4-0} \cdot \left(-\frac{1}{2}u\right)^0 = t^8\] \[4 \cdot (t^{2})^{4-1} \cdot \left(-\frac{1}{2}u\right)^1 = -4t^6\cdot \frac{1}{2} u = -2t^6u\] \[6 \cdot (t^{2})^{4-2} \cdot \left(-\frac{1}{2}u\right)^2 = 6t^4 \cdot \frac{1}{4} u^2 = \frac{3}{2}t^4u^2\] \[4 \cdot (t^{2})^{4-3} \cdot \left(-\frac{1}{2}u\right)^3 = 4t^2 \cdot -\frac{1}{8} u^3 = -\frac{1}{2} t^2u^3\] \[1 \cdot (t^{2})^{4-4} \cdot \left(-\frac{1}{2}u\right)^4 = \frac{1}{16}u^4\]

Combine the terms

Combining all terms, we have the expanded expression: \[\left(t^2-\frac{1}{2}u\right)^4 = t^8 - 2t^6u + \frac{3}{2}t^4u^2 - \frac{1}{2}t^2u^3 + \frac{1}{16}u^4\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Expansion

Algebraic expansion is a technique used to simplify expressions by breaking them down into simpler terms. This often involves expressions raised to a power, like $(t^2 - \frac{1}{2}u)^4$. The goal is to expand these expressions into a sum of terms, each easier to manage. In the case of binomial expressions, the binomial theorem serves as a powerful tool to achieve this expansion. By systematically applying the theorem, we convert a compact expression into a standard polynomial form.

Start with identifying the base expression.
Then, apply the expansion method to break it down.
The result is a series of terms that comprise the simplified form of the expression.

The expanded form offers clear insight into the contribution of each component of the original expression.

Binomial Coefficients

Binomial coefficients are numbers that occur in the process of expanding a binomial raised to a power. These coefficients can be found in Pascal's Triangle or calculated using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where $n$ is the power and $k$ is the term number in the expansion. In the expansion $ (t^{2} - \frac{1}{2}u)^4 $, these coefficients provide the weights for each term in the expansion:

$ \binom{4}{0} = 1 $
$ \binom{4}{1} = 4 $
$ \binom{4}{2} = 6 $
$ \binom{4}{3} = 4 $
$ \binom{4}{4} = 1 $

Each coefficient multiplies the corresponding binomial term, determining its contribution to the overall expansion.

Polynomial Expressions

A polynomial expression is a sum of multiple terms, each consisting of a constant coefficient multiplied by a variable raised to a power. Expanding $(t^2 - \frac{1}{2}u)^4$ results in a polynomial expression where each term involves different powers of $t^2$ and $-\frac{1}{2}u$. Here's how the polynomial expansion looks:

$ t^8 $
$ -2t^6u $
$ \frac{3}{2}t^4u^2 $
$ -\frac{1}{2}t^2u^3 $
$ \frac{1}{16}u^4 $

Each of these terms represents a particular combination of powers from the original expression, showcasing how variables interact in the multiplied form.

Mathematical Calculation

Mathematical calculation involves performing various steps to achieve accurate results from an expression or problem. Let's break it down with the binomial expansion:

Use the binomial theorem to formulate the $n$th power of the expression.
Determine each binomial coefficient accordingly.
Substitute these coefficients and expressions into the expansion formula.
Carefully calculate each term. Remember to handle signs and fractions appropriately.
Finally, sum up all terms to complete the expansion.

For the expression $(t^2 - \frac{1}{2}u)^4$, this calculation involves each term individually considered, utilizing skills in multiplication, exponentiation, and basic fraction handling. This structured approach ensures clarity and precision in obtaining the final expanded form.

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91影视

Use the binomial theorem to expand each expression. $$\left(t^{2}-\frac{1}{2} u\right)^{4}$$

Short Answer

Step by step solution

Identify the expression to expand

Apply the binomial theorem

Calculate the binomial coefficients and expand each term

Combine the terms

Key Concepts

Algebraic Expansion

Binomial Coefficients

Polynomial Expressions

Mathematical Calculation

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