/*! 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 3 Classify the following equations... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 3

Classify the following equations in terms of their degree. $$ 25 y^{3}+y=9 y^{2}-17 y+4 $$

Short Answer

Expert verified
Answer: The degree of the given equation is 3.

Step by step solution

01

Identify terms and their degrees

First, let's rewrite the equation so it's equal to zero: $$ 25y^3 + y - 9y^2 + 17y - 4 = 0 $$ Now, let's identify the term and the degree of each term: - \(25y^3\): degree 3 - \(-9y^2\): degree 2 - \(1y\): degree 1 - \(17y\): degree 1 - \(-4\): degree 0
02

Determine the highest degree

Now that we have identified the degree of each term, we can find the highest degree in the equation. In this case, the highest degree is 3, which corresponds to the term \(25y^3\).
03

Classify the equation

The equation$$ 25 y^{3}+y=9 y^{2}-17 y+4 $$ has a highest degree of 3. Therefore, the equation is classified as a cubic equation.

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 Degree
The degree of a polynomial is a fundamental concept that helps us understand the nature of an equation. The degree of a polynomial is determined by the largest exponent of the variable present in the equation. In a polynomial such as \(25y^3 + y - 9y^2 + 17y - 4 = 0\), to find the degree, we look at each term separately.

  • The term \(25y^3\) has an exponent of 3.
  • The term \(-9y^2\) has an exponent of 2.
  • The terms \(y\) and \(17y\) have exponents of 1.
  • The constant term \(-4\) can be thought of as having an exponent of 0, since \(y^0 = 1\).

Among these terms, the highest exponent is 3. This makes the polynomial a third-degree polynomial.
Understanding the degree of a polynomial is crucial because it indicates the number of roots the polynomial might have and helps us predict the general shape of its graph.
Equation Classification
Classifying an equation based on its degree provides insight into its complexity and the possible methods needed to solve it. The degree directly influences the equation鈥檚 classification.

  • A polynomial with a degree of 0 is just a constant and doesn鈥檛 involve variables.
  • A degree of 1 results in a linear equation, which graphs as a straight line.
  • Degree 2 is known as a quadratic, which forms a parabolic curve on a graph.
  • For degree 3, we have a cubic equation, as in the example \(25y^3 + y - 9y^2 + 17y = 4\).

Cubic equations typically form an S-shaped curve when plotted and may have up to three real roots.

Recognizing an equation鈥檚 classification through its degree helps choose the appropriate strategy for solving it and understanding its graphical behavior.
Algebraic Terms
Understanding algebraic terms is key to working effectively with any equation. An algebraic term consists of several components:
  • Coefficient: The numerical factor in a term, like 25 in \(25y^3\).
  • Variable: The letter representing an unknown value, such as \(y\).
  • Exponent: The power to which the variable is raised, indicating how many times the variable is multiplied by itself, like 3 in \(25y^3\).

In more complex polynomials, terms can be both positive and negative, affecting the equation鈥檚 behavior.
  • The term \(25y^3\) is positive, contributing to the direction the curve takes when plotted.
  • The term \(-9y^2\) is negative, subtracting from the curve in the opposite direction.
  • Constant terms like \(-4\) shift the graph vertically up or down.

Recognizing each component of algebraic terms allows for manipulating and simplifying equations, which is essential for solving them.

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

List, if any should appear, the common factors for the following problems. $$ 6(a+4)+12(a+4) $$

For the following problems, simplify each of the algebraic expressions. $$ 3 h[2 h+5(h+2)] $$

For the following problems, perform the multiplications and combine any like terms. $$ (x-5)^{2} $$

For the following problems, simplify each of the algebraic expressions. $$ 1\left(2 x-6 b+6 a^{2} b+8 b^{2}\right)+1\left(5 x+2 b-3 a^{2} b\right) $$

For the following problems, find the products. $$ \left(a+\frac{1}{2}\right)^{2} $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Logic and Functions

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.