AP Precalculus Section 1.5 Example: The Degree of a Polynomial from a Table

Random AP Precalculus Problems (I found on the Internet). These are not official AP Collegeboard examples, but they will definitely get the job done!

Certainly! Determining the degree of a polynomial from a table of values involves observing the patterns in the rates of change (differences between consecutive y-values). Here's how you can do it:

### Steps to Determine the Degree of a Polynomial from a Table:

1. **Calculate Differences Between Consecutive y-Values**:
- Given a table of values for \(x\) and \(y\) (or \(f(x)\)), compute the differences between consecutive \(y\)-values.

2. **Observe the Rate of Change**:
- For a polynomial of degree \(n\), the \(n\)th differences will be constant.

3. **Identify the Degree**:
- The degree of the polynomial corresponds to the order of the last non-zero constant \(n\)th difference.

### Example:

Consider the following table:

\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
1 & 3 \\
2 & 7 \\
3 & 16 \\
4 & 29 \\
5 & 48 \\
\hline
\end{array}
\]

1. **Calculate First Differences**:
- First differences: \(4, 9, 13, 19\)
2. **Calculate Second Differences**:
- Second differences: \(5, 4, 6\)
3. **Calculate Third Differences**:
- Third differences: \(1, 2\)
4. **Identify the Degree**:
- The third differences are constant (\(1, 2\)). The last non-zero constant difference indicates the degree.
- Since the third differences are constant and the last non-zero difference is the second one, the polynomial is of degree 3 (cubic).

### Interpretation:
- In this case, observing that the third differences are constant and the last non-zero constant difference is the second one implies that the polynomial from the table is likely a cubic polynomial (\(n = 3\)).

By examining the pattern of differences and finding the order of the last non-zero constant difference, you can determine the degree of the polynomial represented by the given table of values.

The Topics covered in AP Precalculus are...

1.1 Change in Tandem
1.2 Rates of Change
1.3 Rates of Change in Linear and Quadratic Functions
1.4 Polynomial Functions and Rates of Change
1.5 Polynomial Functions and Complex Zeros
1.6 Polynomial Functions and End Behavior
1.7 Rational Functions and End Behavior
1.8 Rational Functions and Zeros
1.9 Rational Functions and Vertical Asymptotes
1.10 Rational Functions and Holes
1.11 Equivalent Representations of Polynomial and Rational Expressions
1.12 Transformations of Functions
1.13 Function Model Selection and Assumption Articulation
1.14 Function Model Construction and Application
2.1 Change in Arithmetic and Geometric Sequences
2.2 Change in Linear and Exponential Functions
2.3 Exponential Functions
2.4 Exponential Function Manipulation
2.5 Exponential Function Context and Data Modeling
2.6 Competing Function Model Validation
2.7 Composition of Functions
2.8 Inverse Functions
2.9 Logarithmic Expressions
2.10 Inverses of Exponential Functions
2.11 Logarithmic Functions
2.12 Logarithmic Function Manipulation
2.13 Exponential and Logarithmic Equations and Inequalities
2.14 Logarithmic Function Context and Data Modeling
2.15 Semi-log Plots
3.1 Periodic Phenomena
3.2 Sine, Cosine, and Tangent
3.3 Sine and Cosine Function Values
3.4 Sine and Cosine Function Graphs
3.5 Sinusoidal Functions
3.6 Sinusoidal Function Transformations
3.7 Sinusoidal Function Context and Data Modeling
3.8 The Tangent Function
3.9 Inverse Trigonometric Functions
3.10 Trigonometric Equations and Inequalities
3.11 The Secant, Cosecant, and Cotangent Functions
3.12 Equivalent Representations of Trigonometric Functions
3.13 Trigonometry and Polar Coordinates
3.14 Polar Function Graphs
3.15 Rates of Change in Polar Functions

I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out:

/ nickperich

Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

#APPrecalculus
#PreCalcProblems
#MathMinds
#AdvancedPreCalc
#TrigTales
#PrecalcPuzzles
#FunctionFiesta
#GraphGoals
#CalcReady
#PreCalcLife
#AlgebraicAdventures
#DerivativeDreams
#IntegrationInsights
#MathematicsMagic
#PreCalcReview
#PrecalcConcepts
#LogarithmLove
#TrigonometryTips
#MathMastermind
#APCalcPrep
#Mathematics
#MathMinds
#Math
#Maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study