Maths1_Week4_Polynomials

Maths 1 Week 4: Algebra of Polynomials

0. Prerequisites

NOTE

What you need to know:

• Exponents: $x^2\cdot x^3=x^5$.
• Distributive Property: $a(b+c)=ab+ac$.
• Long Division: Basic division of numbers (e.g., $15÷4=3$ rem $3$).

Quick Refresher

• Monomial: Single term ($3x^2$).
• Binomial: Two terms ($x+1$).
• Polynomial: Many terms ($a_n{x}^n+\cdots +a_0$).

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1. Core Concepts

1.1 Polynomial Features

• Degree ($n$): The highest power of $x$.
  • $n=1$: Linear (Line).
  • $n=2$: Quadratic (Parabola).
  • $n=3$: Cubic (S-shape).
• Leading Coefficient ($a_n$): The number in front of the highest power.
• Turning Points: Places where the graph changes direction (up to down or vice versa).
  • Max Turning Points = $n-1$.

1.2 End Behavior (The “Arms”)

What happens as $x\to \infty$ (Right) and $x\to -\infty$ (Left)?

• Even Degree (Like $x^2$): Both arms go same direction.
  • $a_n>0$: Up/Up ($\nwarrow \nearrow$).
  • $a_n<0$: Down/Down ($\swarrow \searrow$).
• Odd Degree (Like $x^3$): Arms go opposite directions.
  • $a_n>0$: Down/Up ($\swarrow \nearrow$).
  • $a_n<0$: Up/Down ($\nwarrow \searrow$).

1.3 Roots & Multiplicity

• Root ($r$): Where graph crosses X-axis ($P(r)=0$).
• Multiplicity ($k$): How many times a factor $(x-r)$ appears.
  • Odd $k$: Graph Crosses the axis.
  • Even $k$: Graph Touches/Bounces off the axis.

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2. Pattern Analysis & Goated Solutions

Pattern 1: Constructing Polynomials from Roots

Context: “Find a polynomial of degree 3 with roots 1, -2, 3 passing through (0, 6).”

TIP

Mental Algorithm:

1. Template: Write $P(x)=A(x-r_1)(x-r_2)\dots$
2. Plug Roots: Substitute the given roots $r_i$.
3. Find A: Use the extra point $(x,y)$ to solve for the scaling factor $A$.
4. Expand: Multiply out if required (usually factored form is fine unless asked).

Example (Detailed Solution)

Problem: Find a cubic polynomial with roots $-1,2,4$ passing through $(0,8)$. Solution:

1. Template: $P(x)=A(x-r_1)(x-r_2)(x-r_3)$.
2. Plug Roots:
   • $r_1=-1⟹(x-(-1))=(x+1)$.
   • $r_2=2⟹(x-2)$.
   • $r_3=4⟹(x-4)$.
   • $P(x)=A(x+1)(x-2)(x-4)$.
3. Find A:
   • Use point $(0,8)$. Plug $x=0,P(x)=8$.
   • $8=A(0+1)(0-2)(0-4)$.
   • $8=A(1)(-2)(-4)$.
   • $8=8A$.
   • $A=1$.
4. Final Equation: $P(x)=1(x+1)(x-2)(x-4)$. Answer: $P(x)=(x+1)(x-2)(x-4)$.

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Pattern 2: Sign Analysis (Inequalities)

Context: “Solve $(x-1)(x+2)>0$.”

TIP

Mental Algorithm (Wavy Curve Method):

1. Roots: Find all roots.
2. Number Line: Mark roots on line.
3. Test: Pick a huge number (like 100). Is the result positive? Start drawing the wave from the right.
4. Bounce/Cross:
   • Odd Multiplicity $\to$ Cross line (Change sign).
   • Even Multiplicity $\to$ Bounce (Keep sign).

Example (Detailed Solution)

Problem: Solve $(x-3)(x+1{)}^2<0$. Solution:

1. Roots:
   • $x=3$ (Multiplicity 1 - Odd).
   • $x=-1$ (Multiplicity 2 - Even).
2. Number Line: Mark -1 and 3.
3. Right Side Test:
   • Pick $x=100$. $(100-3)(100+1{)}^2=(+)\cdot (+)=(+)$.
   • Region $x>3$ is Positive.
4. Move Left:
   • At $x=3$ (Odd mult): Cross. Region $(-1,3)$ is Negative.
   • At $x=-1$ (Even mult): Bounce. Region $(-\infty ,-1)$ is Negative.
5. Goal: We want $<0$ (Negative regions).
   • This is $(-\infty ,-1)\cup (-1,3)$.
   • Note: We exclude -1 because at -1 the value is 0, not $<0$. Answer: $x\in (-\infty ,-1)\cup (-1,3)$.

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Pattern 3: Division & Remainder

Context: “Find remainder when $P(x)$ is divided by $(x-c)$.”

TIP

Mental Algorithm: Remainder Theorem: The remainder of $P(x)÷(x-c)$ is simply $P(c)$. No need to do long division! Just plug in $c$.

Example (Detailed Solution)

Problem: Find remainder when $x^3-2x^2+x+5$ is divided by $x-2$. Solution:

1. Identify Divisor: $x-2⟹c=2$.
2. Plug in: Calculate $P(2)$.
   • $P(2)=(2{)}^3-2(2{)}^2+(2)+5$.
   • $P(2)=8-2(4)+2+5$.
   • $P(2)=8-8+7$.
   • $P(2)=7$. Answer: Remainder is 7.

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3. Practice Exercises

1. End Behavior: Describe end behavior of $y=-2x^4+x$.
   • Hint: Even degree, Negative coeff. Down/Down.
2. Construction: Find quadratic with roots 2, 5 passing through (3, -2).
   • Hint: $y=A(x-2)(x-5)$. Plug (3, -2) to find A.
3. Inequality: Solve $x(x-2)>0$.
   • Hint: Roots 0, 2. Test regions. $(-\infty ,0)\cup (2,\infty )$.

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🧠 Level Up: Advanced Practice

Question 1: Turning Points & Intervals

Problem: Find intervals where $P(x)=x^3-3x^2-9x+5$ is increasing. Logic:

1. Derivative: $P^{\prime }(x)=3x^2-6x-9$.
2. Critical Points: Set $P^{\prime }(x)=0$.
   • $3(x^2-2x-3)=0⟹3(x-3)(x+1)=0$.
   • $x=3,x=-1$.
3. Test Intervals:
   • $(-\infty ,-1)$: Test $x=-2$. $P^{\prime }(-2)=3(4)-6(-2)-9=12+12-9>0$ (Increasing).
   • $(-1,3)$: Test $x=0$. $P^{\prime }(0)=-9<0$ (Decreasing).
   • $(3,\infty )$: Test $x=4$. $P^{\prime }(4)>0$ (Increasing). Answer: Increasing on $(-\infty ,-1]\cup [3,\infty )$.

Question 2: Polynomial Construction

Problem: Find cubic polynomial with roots 1, 2, 3 passing through $(0,-6)$. Logic:

1. Form: $P(x)=k(x-1)(x-2)(x-3)$.
2. Use Point: $P(0)=-6$.
   • $k(0-1)(0-2)(0-3)=-6$.
   • $k(-1)(-2)(-3)=-6$.
   • $-6k=-6⟹k=1$.
3. Expand: $P(x)=1(x^2-3x+2)(x-3)=x^3-3x^2-3x^2+9x+2x-6$. Answer: $P(x)=x^3-6x^2+11x-6$.

Question 3: Asymptotic Behavior

Problem: As $x\to \infty$, which term dominates in $P(x)=2x^4-100x^3$? Logic:

1. Rule: The term with the highest power dominates.
2. Check: At $x=1000$, $2x^4=2\cdot 10^{12}$, $100x^3=10^{11}$.
3. Result: $2x^4$ is much larger. Answer: $2x^4$.