Maths Week 4: Polynomial Functions

This week, we move beyond linear and quadratic functions to the broader category of Polynomials. These functions are incredibly versatile and are used to model more complex relationships in data, engineering, and science. We’ll focus on understanding their graphs, behavior, and key characteristics.

📚 Table of Contents

1. Fundamental Concepts
2. Question Pattern Analysis
3. Detailed Solutions by Pattern
4. Practice Exercises
5. Visual Learning: Mermaid Diagrams
6. Common Pitfalls & Traps
7. Quick Refresher Handbook

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1. Fundamental Concepts

🎯 What is a Polynomial?

A polynomial function has the general form: $P(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$ where the coefficients ($a_n,a_{n-1},\dots$) are real numbers and the exponents are non-negative integers.

• Degree: The highest exponent, $n$.
• Leading Term: The term with the highest exponent, $a_n{x}^n$.
• Leading Coefficient: The coefficient of the leading term, $a_n$.

📈 Graphing Polynomials: Key Features

1. Roots (x-intercepts) and Multiplicity

• Root: A value of $x$ for which $P(x)=0$. These are the points where the graph crosses or touches the x-axis.
• Multiplicity: The number of times a particular factor $(x-r)$ appears in the factored form of the polynomial.
  • Odd Multiplicity (e.g., $(x-r{)}^1,(x-r{)}^3$): The graph crosses the x-axis at $x=r$.
  • Even Multiplicity (e.g., $(x-r{)}^2,(x-r{)}^4$): The graph touches the x-axis at $x=r$ and “bounces” back. This point is also a turning point.

2. End Behavior

End behavior describes what the function does as $x$ approaches positive or negative infinity ($x\to \infty$ or $x\to -\infty$). It is determined only by the leading term ($a_n{x}^n$).

	Even Degree (n)	Odd Degree (n)
$a_n>0$ (Positive)	Up on both sides $(\nwarrow \cdots \nearrow )$	Down on left, Up on right $(\swarrow \cdots \nearrow )$
$a_n<0$ (Negative)	Down on both sides $(\swarrow \cdots \searrow )$	Up on left, Down on right $(\nwarrow \cdots \searrow )$

3. Turning Points

• Definition: Points where the function changes from increasing to decreasing (a local maximum) or from decreasing to increasing (a local minimum).
• Rule: A polynomial of degree $n$ has at most $n-1$ turning points. For example, a degree 5 polynomial can have at most 4 turning points.

4. Increasing and Decreasing Intervals

The turning points and roots divide the x-axis into intervals. Within each interval, the function is either strictly increasing or strictly decreasing. The behavior can be determined by testing a point in the interval or by observing the graph’s path between roots.

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2. Question Pattern Analysis

From the Week_4_Graded_Assignment, we can identify these key problem patterns:

🎯 Pattern 1: Graph Behavior from Factored Form

• Frequency: High
• Description: Given a polynomial in factored form, determine its behavior (increasing/decreasing, turning points) within specific intervals or at its roots.
• Example: Questions 2, 3, 10.

🎯 Pattern 2: End Behavior & Turning Points

• Frequency: High
• Description: Questions that test your understanding of end behavior (as $x\to \infty$) and the maximum number of turning points based on the polynomial’s degree.
• Example: Questions 1, 8.

🎯 Pattern 3: Finding and Using Roots

• Frequency: Medium
• Description: Problems that require you to find the roots (x-intercepts) of a polynomial and then use them in some calculation (e.g., find their sum).
• Example: Question 4.

🎯 Pattern 4: Polynomial Algebra and Construction

• Frequency: Medium
• Description: Problems involving algebraic manipulation (addition, multiplication) of polynomials or constructing a polynomial based on given properties like its roots and behavior.
• Example: Questions 6, 7.

🎯 Pattern 5: Intersection of Polynomials

• Frequency: Medium
• Description: Finding the intersection points of two polynomial curves. This is solved by setting their equations equal to each other.
• Example: Question 5.

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3. Detailed Solutions by Pattern

🎯 Pattern 1: Graph Behavior from Factored Form

Example (from Graded Assignment Q3):

Consider $p(x)=-\left(x^2-16\right){\left(x-3\right)}^2{\left(2-x\right)}^2\left(x+9\right)$. Choose the correct options. (Two of the options are: “Total number of turning points are 6” and ”$p(x)$ first increases then decreases in the interval $(2,3)$“)

Step-by-Step Solution:

1. Analyze the Factored Form: First, let’s clean up the expression and find the roots and their multiplicities.

   • $x^2-16=(x-4)(x+4)$ (Roots: 4, -4; Multiplicity: 1 each)
   • $(x-3{)}^2$ (Root: 3; Multiplicity: 2 - even)
   • $(2-x{)}^2=(-(x-2){)}^2=(x-2{)}^2$ (Root: 2; Multiplicity: 2 - even)
   • $(x+9)$ (Root: -9; Multiplicity: 1 - odd) The full expression is: $p(x)=-(x-4)(x+4)(x-3{)}^2(x-2{)}^2(x+9)$.

2. Determine the Degree and Leading Coefficient:

   • To find the degree, sum the multiplicities: $1+1+2+2+1=7$. The degree is 7 (odd).
   • The leading coefficient is the product of the coefficients of the highest power of $x$ in each factor, including the initial minus sign: $-(1)(1)(1)(1)(1)=-1$ (negative).

3. Analyze Turning Points:

   • A polynomial of degree $n=7$ has at most $n-1=6$ turning points.
   • The roots with even multiplicity (at $x=2$ and $x=3$) are guaranteed turning points.
   • The graph will have additional turning points between the other roots. The total number of turning points is 6. So, “Total number of turning points of $p(x)$ are 6” is correct.

4. Analyze Behavior in the Interval (2, 3):

   • The roots are at $x=2$ and $x=3$.
   • At $x=2$, the multiplicity is 2 (even), so the graph touches the x-axis and turns around.
   • At $x=3$, the multiplicity is 2 (even), so the graph touches the x-axis and turns around.
   • Let’s check the sign of $p(x)$ in the interval $(2,3)$. Pick a test point, say $x=2.5$. $p(2.5)=-(2.5^2-16)(2.5-3{)}^2(2-2.5{)}^2(2.5+9)$ $p(2.5)=-(\text{neg})(\text{pos})(\text{pos})(\text{pos})=\text{positive}$.
   • Since the graph is positive between 2 and 3, and it must touch the axis at both ends, it must go up from $x=2$ to a local maximum and then come back down to $x=3$.
   • Therefore, in the interval $(2,3)$, the function first increases, then decreases. This statement is correct.

🎯 Pattern 2: End Behavior & Turning Points

Example (from Graded Assignment Q8):

Consider $p(x)=-x^5+5x^4-7x-2$. Which options are true? (Options include end behavior and number of turning points).

Step-by-Step Solution:

1. Identify Degree and Leading Coefficient:

   • The term with the highest power is $-x^5$.
   • Degree (n): 5 (Odd)
   • Leading Coefficient ($a_n$): -1 (Negative)

2. Determine End Behavior:

   • We have an Odd Degree and a Negative Leading Coefficient.
   • Consulting the end behavior rules: the graph goes up on the left and down on the right.
   • This means as $x\to \infty$, $p(x)\to -\infty$. The statement ”$p(x)\to -\infty$ as $x\to \infty$” is correct.

3. Determine Maximum Number of Turning Points:

   • For a polynomial of degree $n=5$, the maximum number of turning points is $n-1=4$.
   • The statement ”$p(x)$ has at most 4 turning points” is correct.

🎯 Pattern 4: Polynomial Algebra and Construction

Example (from Graded Assignment Q7):

$P(x)=x^4+4x^3+x+10$ and $Q(x)=x^3+2x^2-6$. Line $M(x)$ passes through $(2,Q(2))$ with slope 3. Find $P(x)+M(x)Q(x)$.

Step-by-Step Solution:

1. Find the point for the line M(x): We need the point $(2,Q(2))$. $Q(2)=(2{)}^3+2(2{)}^2-6=8+2(4)-6=8+8-6=10$ So the point is $(2,10)$.

2. Find the equation for the line M(x): Use the point-slope form $y-y_1=m(x-x_1)$ with $m=3$ and $(x_1,y_1)=(2,10)$. $M(x)-10=3(x-2)$ $M(x)=3x-6+10$ $M(x)=3x+4$

3. Calculate the expression $M(x)Q(x)$: $(3x+4)(x^3+2x^2-6)$ $=3x(x^3+2x^2-6)+4(x^3+2x^2-6)$ $=(3x^4+6x^3-18x)+(4x^3+8x^2-24)$ $=3x^4+10x^3+8x^2-18x-24$

4. Calculate the final expression $P(x)+M(x)Q(x)$: $(x^4+4x^3+x+10)+(3x^4+10x^3+8x^2-18x-24)$ Combine like terms: $(1+3)x^4+(4+10)x^3+8x^2+(1-18)x+(10-24)$ $=4x^4+14x^3+8x^2-17x-14$

Answer: $4x^4+14x^3+8x^2-17x-14$

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4. Practice Exercises

Exercise 1: End Behavior and Roots

A polynomial $P(x)$ has roots at $x=-2$ (multiplicity 1), $x=1$ (multiplicity 2), and $x=4$ (multiplicity 1). The y-intercept is $P(0)=-8$. a) Write a possible equation for $P(x)$. b) Describe its end behavior.

Hint

The general form will be $P(x) = a(x+2)(x-1)^2(x-4)$. Use the y-intercept to solve for $a$. Then determine the degree and the sign of $a$ to find the end behavior.

Solution

a) **Equation:** $$ P(x) = a(x+2)(x-1)^2(x-4) $$ Use $P(0)=-8$: $$ -8 = a(0+2)(0-1)^2(0-4) $$ $$ -8 = a(2)(1)(-4) = -8a $$ $$ a = 1 $$ So, the equation is $P(x) = (x+2)(x-1)^2(x-4)$.

b) End Behavior: The degree is $1+2+1=4$ (Even). The leading coefficient is $a=1$ (Positive). For an even degree and positive leading coefficient, the graph goes up on the left and up on the right. ($x\to \infty ,P(x)\to \infty$ and $x\to -\infty ,P(x)\to \infty$).

Exercise 2: Analyzing Behavior

For the polynomial $f(x)=-(x+1{)}^3(x-2{)}^2$, describe the behavior of the graph at its roots and determine the end behavior.

Hint

Check the multiplicity of each root to see if the graph crosses or touches. Find the degree and leading coefficient to determine end behavior.

Solution

1. **Behavior at Roots:** * At $x=-1$, the multiplicity is 3 (Odd). The graph **crosses** the x-axis. * At $x=2$, the multiplicity is 2 (Even). The graph **touches** the x-axis and turns around. 2. **End Behavior:** * Degree: $3+2=5$ (Odd). * Leading coefficient: The term with the highest power would be $-(x^3)(x^2) = -x^5$. So, the leading coefficient is -1 (Negative). * For an odd degree and negative leading coefficient, the graph goes **up on the left** and **down on the right**.

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5. Visual Learning: Mermaid Diagrams

End Behavior Decision Tree

graph TD
    A[Start: Look at Leading Term a_n*x^n] --> B{Is degree 'n' Even or Odd?};
    B -- Even --> C{Is a_n > 0?};
    B -- Odd --> D{Is a_n > 0?};
    C -- Yes --> E[Up on Left, Up on Right];
    C -- No --> F[Down on Left, Down on Right];
    D -- Yes --> G[Down on Left, Up on Right];
    D -- No --> H[Up on Left, Down on Right];

Root Multiplicity Behavior

graph LR
    A[Root 'r' from factor (x-r)^m] --> B{Is multiplicity 'm' Odd or Even?};
    B -- Odd --> C[Graph CROSSES x-axis at x=r];
    B -- Even --> D[Graph TOUCHES x-axis at x=r (Turning Point)];

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6. Common Pitfalls & Traps

• Degree vs. Turning Points: A degree $n$ polynomial has at most $n-1$ turning points. It can have fewer (e.g., $y=x^3$ has degree 3 but 0 turning points).
• Negative Signs in Factors: Be careful with factors like $(2-x)$. To determine the leading coefficient, it’s safer to rewrite it as $-(x-2)$. An even power, like $(2-x{)}^2$, becomes $(x-2{)}^2$, while an odd power, $(2-x{)}^3$, becomes $-(x-2{)}^3$.
• Multiplicity vs. Turning Points: While all roots with even multiplicity are turning points, there can be other turning points that are not on the x-axis.
• “At Most” vs. “Exactly”: Remember that the rule for turning points gives an upper limit, not an exact number.

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7. Quick Refresher Handbook

Concept	Rule / Procedure
Degree	Highest power of $x$.
Leading Coefficient	Coefficient of the term with the highest power.
End Behavior Rule	Determined by the degree (even/odd) and leading coefficient (pos/neg).
Roots (x-intercepts)	Set $P(x)=0$ and solve.
Multiplicity Effect	Odd: Crosses x-axis. Even: Touches x-axis (bounces).
Turning Points	At most $n-1$ for a degree $n$ polynomial.
Intersection	Set two polynomial equations equal to each other and solve.
Polynomial Algebra	Combine like terms for addition/subtraction. Use distributive property for multiplication.