Question

Find the polynomial with the smallest degree that goes through the given points. $$(1,5),(-1,3) \text { and }(3,-1)$$

Short answer

The polynomial is \( f(x) = -x^2 + x + 5 \).

Step-by-step solution

Understand the Problem

We need to find a polynomial of minimum degree that passes through the points \((1,5)\), \((-1,3)\), and \((3,-1)\). For three distinct points, a polynomial of degree two (a quadratic) is required.

Set Up the General Form of a Quadratic Polynomial

The general form of a quadratic polynomial is \(f(x) = ax^2 + bx + c\). We need to find the values of \(a\), \(b\), and \(c\) that satisfy the given points.

Substitute the Points into the Equation

Substitute each point into the equation \(f(x) = ax^2 + bx + c\):1. For \( (1,5) \): \( a(1)^2 + b(1) + c = 5 \) results in \( a + b + c = 5 \).2. For \( (-1,3) \): \( a(-1)^2 + b(-1) + c = 3 \) results in \( a - b + c = 3 \).3. For \( (3,-1) \): \( a(3)^2 + b(3) + c = -1 \) results in \( 9a + 3b + c = -1 \).

Solve the System of Equations

Solve the equations:1. \( a + b + c = 5 \)2. \( a - b + c = 3 \)3. \( 9a + 3b + c = -1 \)First, subtract equation (2) from (1):\( (b + b) = 2 \Rightarrow b = 1 \).Now let's eliminate \(b\). Adding equations (1) and (2), we get:\( 2a + 2c = 8 \Rightarrow a + c = 4 \).Substitute \( a + c = 4 \) and \( b = 1 \) into equation (3):\( 9a + 3(1) + c = -1 \Rightarrow 9a + 3 + c = -1 \Rightarrow 9a + c = -4 \).Subtract \( a + c = 4 \) from \( 9a + c = -4 \):\( 8a = -8 \Rightarrow a = -1 \).Substitute \( a = -1 \) into \( a + c = 4 \):\( -1 + c = 4 \Rightarrow c = 5 \).

Write the Final Polynomial

Now that we have \( a = -1 \), \( b = 1 \), and \( c = 5 \), the polynomial is: \( f(x) = -1x^2 + 1x + 5 \) or more simply: \( f(x) = -x^2 + x + 5 \).

Key concepts

Degree of Polynomial

The degree of a polynomial is a fundamental concept in algebra and critical in understanding polynomials.
It tells us the highest power of the variable in the polynomial that is not zero. In simple terms, it's the largest exponent in the polynomial equation.
For example, in the equation \( f(x) = -x^2 + x + 5 \), the degree is 2 because the variable \( x \) has its highest exponent as 2.

Why does the degree matter?

• It determines the basic shape and complexity of the graph.
• The number of solutions the polynomial can have is tied to its degree.
• The number of bends or turns the graph can have is related to its degree.

Understanding the degree of a polynomial helps us in many areas of math, such as solving equations, understanding graph behavior, and interpolation.

System of Equations

A system of equations is a collection of two or more equations with a set of variables.
Solving a system of equations means finding the values of the variables that satisfy every equation in the system simultaneously.
This is crucial when interpolating polynomials as it allows us to find the exact coefficients to match given points.

In our problem, we set up a system of three equations:

• Equation 1: \( a + b + c = 5 \)
• Equation 2: \( a - b + c = 3 \)
• Equation 3: \( 9a + 3b + c = -1 \)

By solving this system, we calculated the precise values of \( a \), \( b \), and \( c \) for the quadratic polynomial.
We used elimination, a method to simplify and solve systems of equations by removing variables step by step.
This straightforward process is key to finding polynomial coefficients that fit a set of points.

Polynomial Interpolation

Polynomial interpolation is a technique to find a polynomial that passes through a particular set of points. It's especially useful in data fitting and numerical analysis.
The process involves determining a polynomial with the smallest possible degree that fits the given data points perfectly.

Why is polynomial interpolation useful?

• It helps in estimating unknown values within a range of known data points.
• It's used in graph plotting to create smooth curves through data points.
• In engineering and science, it aids in creating models from experimental data.

In our specific example, we sought a quadratic polynomial, which means the smallest degree needed is 2, to ensure the polynomial would pass through three given points: \((1,5)\), \((-1,3)\), and \((3,-1)\).
By solving for the coefficients using polynomial interpolation, we get a precise mathematical model that matches our data set perfectly.