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Chapter 5: Problem 99

Find a three-term polynomial of degree 2 whose value will be 1 when it is evaluated for \(x=2\)

Short Answer

Expert verified
The polynomial is \( f(x) = x^2 - 3 \).

Step by step solution

01

Define the General Form of a Quadratic Polynomial

A three-term polynomial of degree 2 can be generally expressed as \( f(x) = ax^2 + bx + c \). This polynomial includes three coefficients \(a\), \(b\), and \(c\), which we need to determine.
02

Plug in the Condition

We know that \( f(2) = 1 \). Substitute \( x = 2 \) into the polynomial equation: \( a(2)^2 + b(2) + c = 1 \). This simplifies to \( 4a + 2b + c = 1 \).
03

Assign Values for Simplicity

Since there are infinitely many solutions, we'll choose convenient values for \(a\) and \(b\) to keep calculations simple. Let's choose \(a = 1\) and \(b = 0\).
04

Solve for c

Substitute \(a = 1\) and \(b = 0\) back into the equation: \(4(1) + 2(0) + c = 1\), which simplifies to \(4 + c = 1\). Solve for \(c\): \( c = 1 - 4 \) or \( c = -3 \).
05

Construct the Polynomial

The polynomial with the values of \(a\), \(b\), and \(c\) is therefore \( f(x) = x^2 - 3 \).

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Key Concepts

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

Polynomial Functions
Polynomial functions are expressions involving variables and coefficients, structured in a specific format using powers of a variable. They are foundational in algebra due to their simplicity and the depth of the mathematical concepts they introduce.
  • **General Form**: A polynomial function can be written in the form: \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), where \( n \) is a non-negative integer, and \( a_n, a_{n-1}, \ldots, a_0 \) are constants known as coefficients.
  • **Types of Polynomials**: Depending on the power, we categorize them as linear (degree 1), quadratic (degree 2), cubic (degree 3), etc.
In our exercise, the polynomial function takes the form of a quadratic polynomial, which means that the highest power of the variable \( x \) is 2. This is why the polynomial can be expressed as \( f(x) = ax^2 + bx + c \). Quadratic polynomials are quite common and serve as excellent models for many natural and mechanical processes.
Degree of a Polynomial
The degree of a polynomial is a crucial aspect that determines its behavior and properties. It is the highest power of the variable present in the polynomial expression.
  • **Understanding Degree**: For example, in the polynomial \( 3x^2 + 5x + 7 \), the degree is 2 because the term with the highest power of \( x \) is \( 3x^2 \).
  • **Impact of Degree**: The degree indicates how many roots the polynomial could potentially have and the number of times a polynomial might intersect the x-axis.
In our problem, we're working with a quadratic polynomial, which has a degree of 2. Quadratic polynomials can have up to two real roots, and their graph is generally a parabola that either opens upwards or downwards depending on the sign of the leading coefficient \( a \). This degree of the polynomial dictates much of how we solve and interpret these expressions.
Coefficients of a Polynomial
Coefficients play a vital role in defining the specific properties of a polynomial. They determine the scale and position of the graph of the polynomial function.
  • **Definition**: Coefficients are the numerical factors multiplied by each term of the polynomial. In \( ax^2 + bx + c \), \( a \), \( b \), and \( c \) are the coefficients.
  • **Role of Each Coefficient**:
    • The coefficient \( a \) (leading coefficient) influences the width and direction of the parabola.
    • The coefficient \( b \) affects the position of the vertex horizontally.
    • The constant term \( c \) represents the y-intercept.
In our exercise, we used specific values for coefficients to construct a polynomial: \( a = 1 \), \( b = 0 \), and \( c = -3 \). This results in the quadratic polynomial \( f(x) = x^2 - 3 \). Knowing how to adjust these coefficients lets us shape the function to meet specific conditions, such as achieving a particular value when evaluated at a specific \( x \).

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