Week 4: Polynomials

1. Polynomial Basics

A polynomial of degree $n$ is: $P(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$

• Degree ($n$): The highest power of $x$. ($a_n\ne 0$).
• Leading Coefficient: $a_n$.
• Constant Term: $a_0$.
• Zero Polynomial: $P(x)=0$ (Degree is undefined).

1.1 Arithmetic of Polynomials

• Addition/Subtraction: Combine like terms.
• Multiplication: Distribute terms. Degree of product = Sum of degrees.
• Division:
  • Division Algorithm: $P(x)=D(x)\cdot Q(x)+R(x)$, where $\mathrm{deg}(R)<\mathrm{deg}(D)$ or $R(x)=0$.
  • Synthetic Division: A shortcut for dividing by $(x-c)$.

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2. Roots and Factors

2.1 Remainder Theorem & Factor Theorem

• Remainder Theorem: When $P(x)$ is divided by $(x-c)$, the remainder is $P(c)$.
• Factor Theorem: $(x-c)$ is a factor of $P(x)$ if and only if $P(c)=0$.

2.2 Finding Roots

For a polynomial of degree $n$:

• It has at most $n$ real roots.
• Complex roots occur in conjugate pairs (if coefficients are real).
• Rational Root Theorem: If $P(x)$ has integer coefficients, any rational root $p/q$ must satisfy:
  • $p$ divides constant term $a_0$.
  • $q$ divides leading coefficient $a_n$.

2.3 Multiplicity

If $P(x)=(x-c{)}^{k}Q(x)$ where $Q(c)\ne 0$:

• $c$ is a root of multiplicity $k$.
• Graph Behavior:
  • Odd $k$: Graph crosses the x-axis at $c$. (e.g., Line, Cubic).
  • Even $k$: Graph touches/bounces off the x-axis at $c$. (e.g., Parabola).

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3. Graphs of Polynomials

Understanding shape without plotting every point.

3.1 End Behavior

Determined by the term with the highest degree ($a_n{x}^n$).

• As $x\to \infty$: Sign depends on $a_n$.
• As $x\to -\infty$:
  • If $n$ is even: Same sign as $x\to \infty$. (Both ends up or both down).
  • If $n$ is odd: Opposite sign. (One end up, one down).

3.2 Turning Points

• A polynomial of degree $n$ has at most $n-1$ turning points (local max/min).

3.3 Graphing Strategy (Goated Procedure)

1. End Behavior: Check leading term.
2. y-intercept: Evaluate $P(0)$.
3. x-intercepts (Roots): Factor $P(x)$. Mark on axis.
4. Multiplicity Check: Decide if crossing or bouncing at roots.
5. Intermediate Value Theorem (IVT): If $P(a)>0$ and $P(b)<0$, there is a root between $a$ and $b$. Useful for estimation.

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4. Goated Examples

Example 1: Constructing Polynomials

Question: Find a polynomial of degree 3 with roots $-1,2,4$ and $y$-intercept $-8$. Solution:

1. Form: $P(x)=a(x+1)(x-2)(x-4)$.
2. Use point $(0,-8)$: $P(0)=a(1)(-2)(-4)=8a$.
3. $8a=-8⟹a=-1$.
4. $P(x)=-1(x+1)(x-2)(x-4)$.

Example 2: Analyzing Graphs

Question: $P(x)=x^3(x-2{)}^2(x+3)$. Describe the graph. Analysis:

1. Degree: $3+2+1=6$ (Even).
2. Leading Coeff: Positive ($+1$).
3. End Behavior: As $x\to \pm \infty ,P(x)\to \infty$ (Up-Up).
4. Roots:
   • $x=0$ (Mult 3, Odd): Crosses (flatter inflection).
   • $x=2$ (Mult 2, Even): Bounces (touches).
   • $x=-3$ (Mult 1, Odd): Crosses.
5. Sketch: Starts high (from $-\infty$), crosses at -3, goes down, crosses at 0, goes up, bounces at 2, goes up to $+\infty$.

Example 3: Division

Question: Divide $P(x)=2x^3-3x^2+4x-5$ by $x-2$. Use synthetic division. Solution:

2 |  2   -3    4   -5
  |       4    2   12
    ------------------
     2    1    6    7

• Quotient: $2x^2+x+6$.
• Remainder: 7.
• Verification: $P(2)=2(8)-3(4)+4(2)-5=16-12+8-5=7$. (Matches!).