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Chapter 8: Problem 1

Fill in the blanks. An equation of the form \(a x^{2}+b x+c=0,\) where \(a \neq 0,\) is called a_________equation.

Short Answer

Expert verified
The equation is called a quadratic equation.

Step by step solution

01

Identify the Given Mathematical Form

The given mathematical form is \(ax^2 + bx + c = 0\). This form includes a quadratic term \(ax^2\), a linear term \(bx\), and a constant term \(c\). The key element here is the term \(ax^2\), which indicates the degree of the polynomial equation.
02

Recall the Definition

An equation is classified according to its highest degree. The highest degree in the equation \(ax^2 + bx + c = 0\) is 2, due to the term \(ax^2\). An equation with a highest degree of 2 is defined as a quadratic equation.
03

Confirm the Condition

The condition given is \(a eq 0\). This ensures that the equation maintains its quadratic nature because if \(a = 0\), the equation becomes linear rather than quadratic. As \(a\) is not zero, the equation is indeed quadratic.

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Key Concepts

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

Polynomial Equation
A polynomial equation is an expression set to a value, typically zero, that is made up of terms consisting of variables raised to various powers and the coefficients. The general form is:
  • The highest power of the variable determines its degree.
  • Each term must consist of a coefficient (number) and the variable raised to a non-negative integer power.
When written in standard form, terms are arranged in descending order of power. For instance, in the quadratic equation form of \(ax^2 + bx + c = 0\), we see a structured polynomial with degrees 2, 1, and 0 respectively for each term. Understanding these components helps in identifying and solving equations systematically.
Degree of Polynomial
The degree of a polynomial refers to the highest power of the variable in the polynomial equation. This determines the 'type' of the polynomial.
  • A polynomial of degree 1 is called linear.
  • A polynomial of degree 2 is termed quadratic.
  • A polynomial of degree 3 is known as cubic, and so forth.
In the equation \(ax^2 + bx + c = 0\), the highest degree is 2, making it a quadratic equation. Understanding the degree is crucial because it tells us the potential number of solutions or "roots" and the basic shape of the graph of the equation.
Linear Term
The linear term in a polynomial is the term where the variable has a power of exactly 1. In the expression \(ax^2 + bx + c\), the linear term is \(bx\). This term impacts the slope or inclination of the polynomial graph at various points.
  • In a linear equation, such as \(bx + c = 0\), this term is even more significant, as it determines the rate at which the graph rises or falls along with the y-intercept, specified by the constant term \(c\).
  • In a quadratic equation, the linear term contributes to the direction and width of the parabola formed by the graph of the equation.
Understanding the role of the linear term is essential for factoring polynomials and graphing equations.
Constant Term
The constant term of a polynomial is the term without any variables, simply representing a constant value. In our equation \(ax^2 + bx + c\), the constant term is \(c\). It doesn't change with the variable, remaining the same no matter the value of the variable.
  • It is important, particularly when analyzing the graph of the polynomial, because it determines where the graph will intersect the y-axis.
  • The value of the constant term is also crucial in determining the sum and product of the roots in polynomial equations via the Vieta's formulas.
Hence, understanding the constant term helps in both graphical and algebraic interpretations of polynomial equations.

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