Problem 25 Use the binomial theorem to expa... [FREE SOLUTION]

91影视

Precalculus: Functions and Graphs

Earl W. Swokowski, Jeffrey A. Cole

$Math Studyset 91影视 Explanations$ Math

12 Edition

Chapter 9: Problem 25

Use the binomial theorem to expand and simplify. $$(3 t-5 s)^{4}$$

Short Answer

Expert verified

The expansion is: $81t^4 - 540t^3s + 1350t^2s^2 - 1500ts^3 + 625s^4$.

Step by step solution

Understanding the Binomial Theorem

The binomial theorem states that \[ (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k \]where $ \binom{n}{k} $ is the binomial coefficient. Here, $a = 3t$, $b = -5s$, and $n = 4$.

Calculate Binomial Coefficients

Calculate the binomial coefficients for $ n = 4 $:\[ \binom{4}{0} = 1, \; \binom{4}{1} = 4, \; \binom{4}{2} = 6, \; \binom{4}{3} = 4, \; \binom{4}{4} = 1 \]These coefficients will multiply each corresponding term in the expansion.

Expand the Expression Using Binomial Coefficients

Using the binomial coefficients, expand \[ (3t - 5s)^4 = \sum_{k=0}^{4} \binom{4}{k} (3t)^{4-k} (-5s)^k \].Calculate each term in the expansion using the coefficients and the powers:- For $ k = 0 $: \[ \binom{4}{0} (3t)^4 (-5s)^0 = 1 \cdot 81t^4 \cdot 1 = 81t^4 \]- For $ k = 1 $: \[ \binom{4}{1} (3t)^3 (-5s)^1 = 4 \cdot 27t^3 \cdot (-5s) = -540t^3s \]- For $ k = 2 $: \[ \binom{4}{2} (3t)^2 (-5s)^2 = 6 \cdot 9t^2 \cdot 25s^2 = 1350t^2s^2 \]- For $ k = 3 $: \[ \binom{4}{3} (3t)^1 (-5s)^3 = 4 \cdot 3t \cdot (-125s^3) = -1500ts^3 \]- For $ k = 4 $: \[ \binom{4}{4} (3t)^0 (-5s)^4 = 1 \cdot 1 \cdot 625s^4 = 625s^4 \]

Write the Final Expanded and Simplified Expression

Combine all the terms from the expansion to get:\[ (3t - 5s)^4 = 81t^4 - 540t^3s + 1350t^2s^2 - 1500ts^3 + 625s^4 \]

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Binomial Coefficients

Binomial coefficients are integral parts of the Binomial Theorem, which allows us to expand expressions raised to a power. In our problem, the expression \[(3t - 5s)^4\]involves the calculation of binomial coefficients for $ n = 4 $. The coefficients \[\binom{4}{k}\]are calculated using the formula:

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

These coefficients arise from Pascal's Triangle or directly through the formula involving factorials, useful for polynomial expansion.

Polynomial Expansion

Polynomial expansion transforms expressions like $(3t - 5s)^4$into a sum of terms using mathematical rules. The Binomial Theorem is pivotal here, stating that:\[(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\]For our given problem:

$a = 3t$
$b = -5s$
$n = 4$
The sum is over $k$ from 0 to the power $n$.

Evaluating each term involves substituting these values with the corresponding binomial coefficients. This results in transforming simple binomial expressions into a series of polynomial terms with different powers.

Algebraic Manipulation

Algebraic manipulation involves combining terms and simplifying expressions, crucial after expanding polynomial terms. From each step in the expansion:

For $k=0$, $ 81t^4 $
For $k=1$, $ -540t^3s $
For $k=2$, $ 1350t^2s^2 $
For $k=3$, $ -1500ts^3 $
For $k=4$, $ 625s^4 $

Combine these results into a single expanded expression:\[81t^4 - 540t^3s + 1350t^2s^2 - 1500ts^3 + 625s^4\]Algebraic manipulation ensures the polynomial is simplified fully, avoiding extra steps and mistakes.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Use the binomial theorem to expand and simplify. $$(3 t-5 s)^{4}$$

Short Answer

Step by step solution

Understanding the Binomial Theorem

Calculate Binomial Coefficients

Expand the Expression Using Binomial Coefficients

Write the Final Expanded and Simplified Expression

Key Concepts

Binomial Coefficients

Polynomial Expansion

Algebraic Manipulation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Applied Mathematics

Decision Maths

Pure Maths

Study anywhere. Anytime. Across all devices.