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Chapter 5: Problem 5

The equation \(x^{3}+x^{2}+x=0\) is not a quadratic equation because ______

Short Answer

Expert verified
The highest power of \(x\) is 3, which makes it a cubic equation.

Step by step solution

01

- Identify the general form of a quadratic equation

A quadratic equation is typically written in the form: \[ ax^{2} + bx + c = 0 \] where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero.
02

- Compare the given equation with the quadratic form

The given equation is: \[ x^{3} + x^{2} + x = 0 \] Notice that the highest degree of the term in this equation is 3 (\( x^{3} \)), while a quadratic equation can have a maximum degree of 2.
03

- Conclude why the given equation is not quadratic

Since the highest power of \(x\) in the equation is 3, this makes it a cubic equation, not a quadratic equation. A quadratic equation must have the highest power of 2.

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Key Concepts

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

Quadratic Equations
A quadratic equation is one of the fundamental concepts in algebra. It is an equation of the form: \[ ax^{2} + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The defining characteristic of a quadratic equation is that the highest degree of the variable \(x\) is 2. Because the degree is 2, these equations produce a parabolic graph when plotted on a coordinate plane. Solving quadratic equations can be done using several methods:
  • Factoring
  • Using the quadratic formula, \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\]
  • Completing the square
  • Graphical methods
These methods are useful for finding the roots or solutions of the quadratic equation, which are the values of \(x\) that make the equation true.
Polynomial Degree
The term 'polynomial degree' refers to the highest power of the variable \(x\) present in a polynomial equation. A polynomial can be classified by its degree:
  • A polynomial of degree 1 is a linear equation (e.g., \(ax + b = 0\)).
  • A polynomial of degree 2 is a quadratic equation (e.g., \(ax^{2} + bx + c = 0\)).
  • A polynomial of degree 3 is a cubic equation (e.g., \(ax^{3} + bx^{2} + cx + d = 0\)).
The degree of a polynomial gives an important clue about the shape of its graph and the number of solutions (roots) it can have. For example, a cubic equation can have up to three real roots because its highest degree term is \(x^3\).
Equation Forms
There are various forms of polynomial equations, each named based on its highest-degree term. Recognizing these forms is crucial for choosing the correct method to solve them. Here are a few common types:
  • Linear Equation: \(ax + b = 0\) - Degree 1

  • Quadratic Equation: \(ax^{2} + bx + c = 0\) - Degree 2

  • Cubic Equation: \(ax^{3} + bx^{2} + cx + d = 0\) - Degree 3

These forms dictate the number of solutions and the complexity of finding them. For example, quadratic equations involve at most two solutions and can be solved using the quadratic formula. On the other hand, cubic equations can be more challenging because they involve higher-degree polynomial terms and can have up to three real roots.

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Most popular questions from this chapter

Factor each binomial completely. \(2 x^{3}-16 y^{3}\)

Solve each problem. A rock is dropped off a cliff that is \(256 \mathrm{ft}\) above a river. The height \(h\) in feet of the rock after \(t\) seconds is given by $$ h=-16 t^{2}+256 $$ After how many seconds does the rock hit the water?

Factor each polynomial completely. \(4 x^{2}+4 x+1-y^{2}\)

Factor each binomial completely. \(27 m^{6}+8 n^{3}\)

Solve each problem. Find three consecutive odd integers such that 3 times the sum of all three is 18 more than the product of the first and second integers.
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