/*! 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 7 For the following exercises, fin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 7

For the following exercises, find the \(x\) - or t-intercepts of the polynomial functions. $$ C(t)=3(t+2)(t-3)(t+5) $$

Short Answer

Expert verified
The \( t \)-intercepts are \( -2, 3, \) and \( -5 \).

Step by step solution

01

Identify the Equation

We start with the polynomial function given by \( C(t) = 3(t+2)(t-3)(t+5) \). Here, \( t \) represents the variable, similar to \( x \) in other contexts.
02

Understanding Intercepts

The \( t \)-intercepts occur where the polynomial is equal to zero, \( C(t) = 0 \). This means finding values of \( t \) that make the equation \( C(t) = 3(t+2)(t-3)(t+5) \) equal to zero.
03

Set the Equation to Zero

To find the intercepts, we set the polynomial equal to zero: \[ 3(t+2)(t-3)(t+5) = 0 \] Since 3 is a nonzero constant, we can ignore it to focus on the factors involving \( t \).
04

Solve for Each Factor

Set each factor equal to zero and solve for \( t \): 1. \( t+2=0 \) leads to \( t=-2 \)2. \( t-3=0 \) leads to \( t=3 \)3. \( t+5=0 \) leads to \( t=-5 \)
05

List the Intercepts

The \( t \)-intercepts of the polynomial are the values of \( t \) calculated in the previous step: \( -2, 3, \) and \( -5 \). These are the points where the function crosses the \( t \)-axis.

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.

X-Intercepts
Understanding x-intercepts is key to analyzing polynomial functions. In simple terms, x-intercepts are the points where the graph of a polynomial crosses the x-axis. Since the y-coordinate at any x-intercept is zero, the x-intercepts are the roots of the equation when the polynomial is set to zero.

To find x-intercepts, follow these steps:
  • Set the polynomial equation equal to zero.
  • Solve for the variable, typically x. This involves finding the values of x that satisfy the equation.
Remember, for a polynomial function, such as one structured like our given polynomial function in terms of t, the x-intercepts are analogous to the t-intercepts. Therefore, learning how to detect these intercepts enhances our understanding of how the polynomial behaves graphically.
T-Intercepts
The concept of t-intercepts is closely related to x-intercepts, but applies when the variable in the polynomial is t instead of x. The t-intercepts occur where the graph of the polynomial crosses the t-axis, meaning the value of the function is zero at these points.

For the function provided, the intercepts can be found by examining where the polynomial equals zero: \( C(t) = 3(t+2)(t-3)(t+5) = 0 \). When this is set to zero, each factor of the polynomial must be zero individually.
  • Solving \( t+2=0 \) gives \( t=-2 \)
  • Solving \( t-3=0 \) gives \( t=3 \)
  • Solving \( t+5=0 \) gives \( t=-5 \)
Thus, the t-intercepts for the polynomial are at t = -2, 3, and -5. These points provide insights into where the polynomial graph interacts with the t-axis.
Factoring Polynomials
Factoring polynomials is a fundamental skill in finding intercepts and simplifying complex equations. It involves breaking down a polynomial into simpler terms (factors), which, when multiplied together, give the original polynomial.

The provided function \( C(t) = 3(t+2)(t-3)(t+5) \) is a perfect illustration of factoring. Here is how factoring helps:
  • When the polynomial is completely factored, as in \( 3(t+2)(t-3)(t+5) \), each binomial factor like \( (t+2) \), \( (t-3) \), and \( (t+5) \) represents potential solutions or zeros.
  • If each factor is set to zero, solving these simpler equations quickly yields the roots or intercepts of the polynomial.
Factoring is not only essential for finding intercepts but also plays a critical role in calculus and higher-level mathematics. It reveals the structure of a polynomial and highlights its zeros, aiding in graph sketching and providing insights into function behavior.

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

For the following exercises, use the given information to find the unknown value. \(y\) varies inversely with the cube root of \(x\). When \(x=27,\) then \(y=5 .\) Find \(y\) when \(x=125 .\)

For the following exercises, use the given information to find the unknown value. \(y\) varies jointly as the square of \(x\) and the square root of \(z\). When \(x=2\) and \(z=9,\) then \(y=24\). Find \(y\) when \(x=3\) and \(z=25\).

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies inversely as the square of \(x\) and when \(x=3, \quad y=2\).

For the following exercises, find the inverse of the function and graph both the function and its inverse. $$ f(x)=x^{2}-6 x+1, \quad x \geq 3 $$

For the following exercises, find the inverse of the functions. $$ f(x)=\frac{3}{x-4} $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

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.