Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 10: Problem 54
Find the indicated term in the expansion of the given expression. Ninth term of \((3-z)^{10}\)
Short Answer
Expert verified
The ninth term is \(405z^8\).
Step by step solution
01
Identify the Binomial Expansion Formula
The expansion of a binomial expression \((a+b)^n\) can be found using the Binomial Theorem which states: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). This will give us each term in the expansion.
02
Set the Variables
In the expression \((3-z)^{10}\), identify \(a = 3\), \(b = -z\), and \(n = 10\). We are looking for the ninth term in the expansion.
03
Determine the Term's Position
The \(k\)-th term in the binomial expansion is \(T_{k+1} = \binom{n}{k} a^{n-k} b^k\). Here, the ninth term corresponds to \(k+1 = 9\), hence \(k = 8\).
04
Substitute Values into the Formula
Using \(n = 10\), \(a = 3\), \(b = -z\), and \(k = 8\), substitute into the formula for the \(k\)-th term: \[ T_9 = \binom{10}{8} (3)^{10-8} (-z)^8 \].
05
Solve for the Binomial Coefficient
Calculate the binomial coefficient: \(\binom{10}{8} = \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45\).
06
Evaluate Power Terms
Calculate the powers: \((3)^2 = 9\) and \((-z)^8 = z^8\), because the even power negates the negative sign.
07
Compute the Resulting Term
Multiply the components: \(T_9 = 45 \times 9 \times z^8 = 405z^8\).
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 Expansion
The binomial expansion is a versatile technique used in algebra to expand expressions of the form \((a+b)^n\). This expression follows a pattern described by the Binomial Theorem. Each term in the expansion has a structure that depends on powers of both components \(a\) and \(b\), raised to decreasing and increasing powers, respectively.
- The first term in any expansion is \(a^n\).
- The second term is \(a^{n-1}b\), and this pattern continues.
- This structure allows easy calculation of each individual term without fully expanding the expression.
Binomial Coefficients
At the heart of binomial expansion lie the binomial coefficients, denoted by \(\binom{n}{k}\). These coefficients determine the weight of each term in the series. For an expression \((a+b)^n\), the coefficient \(\binom{n}{k}\) denotes the coefficient of the term \(a^{n-k}b^k\).
- These coefficients are determined using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
- They represent the number of ways to choose \(k\) elements from a set of \(n\) elements, commonly known as "combinations".
Polynomial Expansion
Polynomial expansion refers to expressing a polynomial in terms of its simplest elements or base components. When expanding a binomial expression like \((3-z)^{10}\), it involves expressing this as a series where each term reflects the structure determined by the binomial theorem.
- Each term is essentially a monomial comprised of constant coefficients and variables raised to a power.
- Through expansion, we transform binomials into a sum of individual monomial terms.
- This makes it easier to handle more complex algebraic manipulations and simplify expressions further.
Exponential Functions
While not directly used in polynomial expansions, exponential functions showcase a broader category of power functions, denoted typically as \(e^x\). They form a crucial part of algebra and calculus, providing models for growth and decay processes in nature and technology.
- Exponential functions involve a constant base (like 2, 10, or \(e\)) raised to the power of a variable.
- They differ from polynomial functions, which involve variables raised to constant powers.
