/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 40 Expand and simplify the rule of ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 40

Expand and simplify the rule of the function \(g .\) $$f(x)=2 x^{3}+3 x^{2}+x \text { and } g(x)=f(x-1)$$

Short Answer

Expert verified
Answer: The expanded and simplified expression for g(x) is g(x) = 2x^3 - 3x^2 + x.

Step by step solution

01

Substituting (x-1) into f(x)

To find the function \(g(x)\), we need to substitute \((x-1)\) into \(f(x)\). That means wherever we see an \(x\) in the function \(f(x)\), we will replace it with \((x-1)\). So, $$g(x) = f(x-1) = 2(x-1)^3 + 3(x-1)^2 + x - 1$$
02

Expanding (x-1) terms

Now, we need to expand the cubes and squares of \((x-1)\) terms. Using the binomial theorem, we can expand \((x-1)^3\) and \((x-1)^2\) as follows: $$(x-1)^3 = x^3 - 3x^2 + 3x - 1$$ $$(x-1)^2 = x^2 - 2x + 1$$
03

Substitute the expanded terms into g(x)

Now that we have the expanded forms of \((x-1)^3\) and \((x-1)^2\), we can substitute them back into the expression for \(g(x)\): $$g(x) = 2(x^3 - 3x^2 + 3x - 1) + 3(x^2 - 2x + 1) + x - 1$$
04

Distribute the constants in g(x)

Now, we need to distribute the constants 2 and 3 in the expression of \(g(x)\). This will give us: $$g(x) = 2x^3 - 6x^2 + 6x - 2 + 3x^2 - 6x + 3 + x - 1$$
05

Combine like terms

Finally, let's combine the like terms in the expression of \(g(x)\) to simplify it: $$g(x) = 2x^3 - 6x^2 + 3x^2 + 6x - 6x + x - 2 + 3 - 1$$ $$g(x) = 2x^3 - 3x^2 + x$$ So, the expanded and simplified expression for \(g(x)\) is: $$g(x) = 2x^3 - 3x^2 + x$$

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.

Polynomial Functions
A polynomial function is a mathematical expression consisting of variables and coefficients, structured in a form where non-negative integer powers of the variable are used. In the function \( f(x) = 2x^3 + 3x^2 + x \), we have a cubic polynomial, which means the highest degree of any term is 3. Each term in a polynomial can be characterized by:
  • its coefficient - for instance, 2 in \(2x^3\),
  • its base - which is the variable \(x\), and
  • its exponent - which represents the power or degree of the term, such as 3 in \(x^3\).
Polynomials are foundational in algebra because they allow for the modeling and solving of real-world problems. They provide both the flexibility required to describe diverse functional relationships and the simplicity needed to be manageable.
Binomial Theorem
The Binomial Theorem is a powerful algebraic tool that provides a method for expanding expressions that are raised to a power, such as \((x - 1)^3\). It states that \((a + b)^n\) can be expanded as:\[\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]This formula incorporates binomial coefficients \(\binom{n}{k}\), which can be found using Pascal's Triangle or calculated as\(\frac{n!}{k!(n-k)!}\). For example, when applying the binomial theorem to expand \((x-1)^3\), we calculate:
  • \(x^3\) \((k=0)\)
  • \(-3x^2\) \((k=1)\)
  • \(+3x\) \((k=2)\)
  • \(-1\) \((k=3)\)
Thus, the expansion becomes \(x^3 - 3x^2 + 3x - 1\), following the calculated sequence.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations that represent mathematical relationships. In this exercise, we worked with the expression \[g(x) = 2x^3 - 6x^2 + 3x^2 + 6x - 6x + x - 2 + 3 - 1\], which was derived during the process of expanding \(f(x-1)\). When simplifying an algebraic expression:
  • Identify like terms - terms that have the same variable part and exponent.
  • Combine like terms by adding or subtracting coefficients.
This simplification process helps to reduce complex expressions into their simplest form, making them easier to understand and manipulate solving problems.
Function Composition
Function composition involves creating a new function by applying one function to the results of another. In our exercise, the function \(g(x)\) is defined as \(f(x-1)\). This is an example of function composition, where we input \((x-1)\) instead of \(x\) into the function \(f\). Function composition allows:
  • Modification of existing functions to shift, scale, or transform them.
  • Building more complex functions from simpler ones.
It's an essential part of higher level mathematics, providing a framework for constructing solutions that build on simpler base functions. The process of substituting \(x-1\) for \(x\) in \(f(x)\) is precisely the means by which function composition is commonly performed.

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

In Exercises \(49-54,\) you are asked to find a geometric sequence. In each case, round the common ratio r to four decimal places. Data from the U.S. Centers for Disease Control and Prevention indicate that the number of newly reported cases of AIDs each year can be approximated by a geometric sequence \(\left\\{a_{n}\right\\},\) where \(n=1\) corresponds to 2000 (a) If there were 40,758 cases reported in 2000 and 41,573 cases reported in \(2001,\) find a formula for \(a_{n}\) (b) About how many cases were reported in \(2004 ?\) (c) Find the total number of cases reported from 2000 to 2007 (inclusive).

In Exercises \(23-30,\) show that the given sequence is geometric and find the common ratio. $$\left\\{5^{n+2}\right\\}$$

In Exercises \(39-42,\) find the kth partial sum of the geometric sequence \(\left\\{a_{n}\right\\}\) with common ratio \(r\). $$k=7, a_{2}=6, r=2$$

In Exercises \(43-48,\) find the sum. $$\sum_{j=1}^{6} 4\left(\frac{3}{2}\right)^{j-1}$$

Deal with prime numbers. A positive integer greater than 1 is prime if its only positive integer factors are itself and 1. For example, 7 is prime because its only factors are 7 and \(1,\) but 15 is not prime because it has factors other than 15 and 1 (namely, 3 and 5 ). Find the first five terms of the sequence. \(a_{n}\) is the number of prime integers less than \(n\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.