Polynomial Anchor Notes Precalculus Reteach Notes
Function Operations & Polynomials
Core Reference Sheet: TEKS P.2B, P.2D, P.2E, P.2F, P.2G, P.2H
Unit Mastery Guide Texas High School Precalculus
1
Function Operations & Composite Domain (TEKS P.2B)
Inner Before Outer
The 2-Step Composite Domain Rule:
For \( (f \circ g)(x) = f(g(x)) \), find values satisfying:
1. \( x \) is in the domain of inner \( g(x) \).
2. Output \( g(x) \) is in the domain of outer \( f \).
⚠️ Rule: Never simplify the composite fraction before identifying inner restrictions!
Worked Anchor Model:
\( f(x) = \frac{1}{x-3}, \quad g(x) = \sqrt{x+2} \)
• Inner domain: \( x+2 \ge 0 \implies x \ge -2 \)
Composite Domain: \( [-2, 7) \cup (7, \infty) \)
2
Synthetic Division & Remainder Theorem (TEKS P.2D, P.2E)
Algorithmic Division
Core Theorems:
• Remainder Theorem: Dividing \( P(x) \) by \( (x - c) \) yields constant remainder \( R = P(c) \).
• Factor Theorem: \( (x - c) \) is an exact factor if and only if \( P(c) = 0 \).
⚠️ Placeholder Rule: Always insert 0 for any missing power terms before setting up coefficients!
Division Setup Example: \( P(2) \)
Divide \( 2x^3 - 5x^2 + 0x + 4 \) by \( (x - 2) \):
2 | 2 -5 0 4
4 -2 -4
2 -1 -2 | 0
Remainder = 0 ⇒ \( (x-2) \) is a factor! Quotient = \( 2x^2 - x - 2 \).
3
Linear Factorization & Rational Roots (TEKS P.2D, P.2F)
Exact Real Roots
Factor-Root Foundations:
• If \( c \) is a real zero, then \( (x - c) \) is a linear factor.
• Rational Root Theorem: Potential rational zeros are \( \pm \frac{p}{q} \) where \( p \) divides constant \( a_0 \) and \( q \) divides leading coefficient \( a_n \).
\( P(x) = a_n(x - c_1)(x - c_2)\dots(x - c_n) \)
Reconstructing Polynomials:
Given integer roots \( x = -3, 1, 4 \) (monic cubic):
\( P(x) = (x + 3)(x - 1)(x - 4) \)
Notice constant term \( 12 = (-3)(1)(4)(-1)^3 \).
Texas Precalculus Remediation Page 1 of 2 Study Notes & Reference Card
Precalculus Reteach Notes
Graphs, Multiplicity & Operations
Core Reference Sheet: TEKS P.2B, P.2F, P.2G, P.2H
Visual Signatures Multiplicity & Limits
4
Roots, Multiplicity & Axis Behaviors (TEKS P.2F, P.2H)
Sign Changes
1
Multiplicity 1
\( (x - c)^1 \)
Crosses Linearly
Passes straight through axis. Sign changes.
2k
Even Multiplicity
\( (x - c)^2, (x - c)^4 \)
Tangent / Bounces
Touches and turns around. No sign change!
2k+1
Odd Multiplicity > 1
\( (x - c)^3, (x - c)^5 \)
Inflects / Wiggles
Flattens horizontally as it crosses. Sign changes.
Model Function: For \( f(x) = -2(x+3)^2(x-1)(x-4)^3 \): Bounce at -3, Cross at 1, Inflect at 4.
5
End Behavior & Formal Limit Notation (TEKS P.2G)
Leading Term Test
End behavior is exclusively determined by the leading term \( a_n x^n \):
Even Degree (\( n \) even): Same Direction
\( a_n > 0 \): \( \lim_{x \to \infty} f(x) = \infty, \; \lim_{x \to -\infty} f(x) = \infty \) (↑ ↑)
\( a_n < 0 \): \( \lim_{x \to \infty} f(x) = -\infty, \; \lim_{x \to -\infty} f(x) = -\infty \) (↓ ↓)
Odd Degree (\( n \) odd): Opposite Directions
\( a_n > 0 \): \( \lim_{x \to \infty} f(x) = \infty, \; \lim_{x \to -\infty} f(x) = -\infty \) (↓ ↑)
\( a_n < 0 \): \( \lim_{x \to \infty} f(x) = -\infty, \; \lim_{x \to -\infty} f(x) = \infty \) (↑ ↓)
6
Difference Quotient & Average Rate of Change (TEKS P.2B)
Slope Protocol
Algebraic Step Protocol:
\( \frac{f(x+h) - f(x)}{h}, \quad h \ne 0 \)
1. Evaluate \( f(x+h) \) by substituting \( (x+h) \) into each \( x \).
Model for \( f(x) = x^2 - 4x + 3 \):
\( f(x+h) = x^2 + 2xh + h^2 - 4x - 4h + 3 \)
Difference Quotient = \( 2x + h - 4 \)
Texas Precalculus Remediation Page 2 of 2 Study Notes & Reference Card
Polynomial Mastery Worksheet Precalculus TEKS Intervention
Function Operations & Dynamics
Texas Standards: P.2B, P.2C | Targeted Remediation Packet
Name:
Date:
Period: Score:
RETEACH ANCHOR: Operations & Composite Domain
• Quotient Domain \( (f/g)(x) \): Intersect domain of \( f \) and domain of \( g \), then exclude any \( x \) where \( g(x) = 0 \).Composite Domain \( (f \circ g)(x) = f(g(x)) \): \( x \) must be in domain of \( g \), AND \( g(x) \) must be in domain of \( f \).Inverse Functions: \( f(g(x)) = x \) and \( g(f(x)) = x \). Domain of \( f \) equals range of \( f^{-1} \).
1. Given \( f(x) = \sqrt{x+4} \) and \( g(x) = 2x - 6 \):
TEKS P.2B
Find the quotient function \( h(x) = \left(\frac{f}{g}\right)(x) \) and state its domain in interval notation.
Algebraic Formulation of \( h(x) \):
Domain Determination (Interval Notation):
2. Given \( f(x) = \frac{3}{x-5} \) and \( g(x) = \frac{2}{x} \):
TEKS P.2B
Determine the composite function \( (f \circ g)(x) = f(g(x)) \) in simplified single-fraction form and establish its restricted domain.
Composite Substitution & Simplification:
Domain Restrictions & Final Interval:
3. Invertibility & Algebraic Verification:
TEKS P.2C
Let \( f(x) = \frac{2x+1}{x-4} \). Find \( f^{-1}(x) \), then verify algebraically that \( f(f^{-1}(x)) = x \).
Find \( f^{-1}(x) \):
Verification Proof \( f(f^{-1}(x)) = x \):
Unit: Function Operations & Inverses Page 1 of 3 Precalculus TEKS Recovery
Part II: Polynomial Structure
Division, Factoring & Rational Zeros
TEKS P.2D, P.2E, P.2F
RETEACH ANCHOR: Synthetic Division & Linear Factors
• Synthetic Division: When dividing by \( (x - c) \), use \( c \) in the box. Supply zeros for missing degree powers.Factor Theorem: \( c \) is a zero if and only if synthetic division remainder \( R = P(c) = 0 \).Linear Factorization: Given roots \( c_1, c_2, \dots, c_n \), polynomial factors into \( P(x) = a(x - c_1)(x - c_2)\dots(x - c_n) \).
4. Synthetic Division with Missing Degree Terms:
TEKS P.2E
Divide \( P(x) = 3x^4 - 2x^2 + 5x - 7 \) by \( (x + 2) \). Write the quotient polynomial \( Q(x) \) and the remainder \( R \).
Synthetic Division Setup & Table: Quotient: \( Q(x) \) | Remainder: \( R \)
5. Complete Factorization & Rational Roots:
TEKS P.2D, P.2F
Given that \( x = 4 \) is a known root of \( P(x) = x^3 - 5x^2 + 2x + 8 \), use synthetic division to find the depressed quadratic quotient, factor completely, and determine all real rational roots.
Depressed Quadratic Quotient:
Complete Factors & All Roots:
6. Polynomial Reconstruction with Integer Roots:
TEKS P.2D
Write a degree-3 monic polynomial \( P(x) \) in standard form with given integer roots \( x = -2 \), \( x = 1 \), and \( x = 4 \).
Linear Factor Product Expansion & Standard Form:
Unit: Polynomial Division & Factors Page 2 of 3 Precalculus TEKS Recovery
Part III: Graph Attributes & Operations
End Behavior, Multiplicity & Operations
TEKS P.2B, P.2F, P.2G, P.2H
RETEACH ANCHOR: Graph Behaviors & Function Operations
• Multiplicities & Limits: Mult 1 crosses linearly; even mult bounces; odd mult > 1 inflects. Leading term \( a_n x^n \) dictates \( \lim_{x \to \pm\infty} f(x) \).Function Operations & Difference Quotient: For \( \frac{f(x+h) - f(x)}{h} \), carefully expand \( f(x+h) \), subtract \( f(x) \), factor out \( h \), and simplify.Polynomial Compositions: To find \( (f \circ g)(x) \), substitute the entire function \( g(x) \) into every input variable of \( f(x) \).
7. Comprehensive Polynomial Analysis:
TEKS P.2F, P.2G, P.2H
Given \( f(x) = -2(x + 3)^2(x - 1)(x - 4)^3 \):
Leading Term: Degree = _____ , Leading Coeff = _____
End Behavior (Limit Notation):
\( \lim_{x \to \infty} f(x) = \) _________
\( \lim_{x \to -\infty} f(x) = \) _________
y-intercept: \( (0, \text{\_\_\_\_\_}) \)
Zero, Multiplicity & Behavior Table:
ZeroMultiplicityAxis Behavior
\( -3 \)
\( 1 \)
\( 4 \)
8. Function Operations & Composite Synthesis:
TEKS P.2B
Given \( f(x) = x^2 - 4x + 3 \) and \( g(x) = 2x - 5 \):
(a) Difference Quotient \( \frac{f(x+h) - f(x)}{h} \) (\( h \ne 0 \)):
(b) Composite Polynomial \( (f \circ g)(x) \) in Standard Form:
TEKS Mastery Self-Check & Confidence Level:
Needs Reteach Progressing Mastered
Teacher Checkpoint Feedback: ____________________________________________________________________
Unit: Graphs & Function Operations Page 3 of 3 Precalculus TEKS Recovery
Polynomial Mastery Key Teacher Key & Intervention Guide Precalculus Remediation
Part I: Function Operations & Inverses
TEKS P.2B, P.2C | Fully Worked Solutions & Error Diagnostics
Mastery Threshold:
80% Algebraic Accuracy
Problem 1 Solution: Quotient Function & Domain P.2B
Algebraic Formulation:
\( h(x) = \frac{f(x)}{g(x)} = \frac{\sqrt{x+4}}{2x-6} = \frac{\sqrt{x+4}}{2(x-3)} \)
Numerator requires \( x + 4 \ge 0 \implies x \ge -4 \).
Domain Restrictions:
Denominator requires \( 2x - 6 \ne 0 \implies x \ne 3 \).
Domain: \( [-4, 3) \cup (3, \infty) \)
⚠️ Diagnostic Error: Students often forget to exclude \( x = 3 \) because the radical is defined at \( x = 3 \). Remind students that denominator zero takes absolute precedence.
Problem 2 Solution: Composite Function & Domain P.2B
Composition & Simplification:
\( f(g(x)) = \frac{3}{\frac{2}{x} - 5} = \frac{3}{\frac{2 - 5x}{x}} = \frac{3x}{2 - 5x} \)
Domain Determination:
1. Inner \( g(x) \): \( x \ne 0 \)
2. Outer denominator: \( 2 - 5x \ne 0 \implies x \ne 2/5 \)
Domain: \( (-\infty, 0) \cup (0, 2/5) \cup (2/5, \infty) \)
⚠️ Diagnostic Error: Students routinely omit \( x \ne 0 \) because the simplified expression \( \frac{3x}{2-5x} \) appears defined at \( x = 0 \). Stress: domain is governed by the unsimplified composite chain!
Problem 3 Solution: Invertibility & Algebraic Proof P.2C
Deriving \( f^{-1}(x) \):
\( x = \frac{2y+1}{y-4} \implies x(y-4) = 2y+1 \)
Verification Proof:
\( f(f^{-1}(x)) = \frac{2\left(\frac{4x+1}{x-2}\right) + 1}{\frac{4x+1}{x-2} - 4} = \frac{\frac{8x+2 + x-2}{x-2}}{\frac{4x+1 - 4x+8}{x-2}} \)
Intervention Prompt: Ask the student: "If a value breaks the inside machine, can it ever reach the outside machine?" This metaphor solidifies why inner restrictions persist.
TEKS Intervention: Function Operations Page 1 of 3 Teacher Solutions Key
Teacher Key & Intervention Guide Precalculus Remediation
Part II: Polynomial Division & Rational Roots
TEKS P.2D, P.2E, P.2F | Algorithmic Steps & Factorizations
Target Standard:
Synthetic Division & Linear Factors
Problem 4 Solution: Synthetic Division with Missing Term P.2E
Divide \( 3x^4 + 0x^3 - 2x^2 + 5x - 7 \) by \( (x + 2) \implies c = -2 \):
-2 | 3 0 -2 5 -7
-6 12 -20 30
3 -6 10 -15 | 23
Quotient \( Q(x) \): \( 3x^3 - 6x^2 + 10x - 15 \)
Remainder \( R \): \( 23 \) (or \( \frac{23}{x+2} \))
⚠️ Error Callout: 85% of student errors occur from omitting the \( 0x^3 \) placeholder. Require students to write degree index numbers above row 1: \( [x^4, x^3, x^2, x^1, x^0] \).
Problem 5 Solution: Factorization & Rational Roots P.2D, P.2F
Depressed Quadratic:
Synthetic division by 4 on \( x^3 - 5x^2 + 2x + 8 \):
\( Q(x) = x^2 - x - 2 \)
Factors easily: \( (x - 2)(x + 1) \)
Complete Factorization & Roots:
\( P(x) = (x - 4)(x - 2)(x + 1) \)
Roots: \( \{-1, 2, 4\} \)
All 3 roots are distinct rational integers.
Problem 6 Solution: Polynomial Reconstruction P.2D
Given integer roots: \( x = -2, \; 1, \; 4 \).
Linear factors = \( (x + 2)(x - 1)(x - 4) \)
Multiply first two factors:
Multiply by third factor \( (x - 4) \):
Scaffolding Tip: Encourage students to organize polynomial multiplication with a 2x3 area grid (box method) to avoid dropped signs when distributing across quadratic trinomials.
TEKS Intervention: Division & Roots Page 2 of 3 Teacher Solutions Key
Teacher Key & Intervention Guide Precalculus Remediation
Part III: Graphs & Function Operations
TEKS P.2B, P.2G, P.2H | Worked Solutions & Intervention Rubric
Spiral Assessment
Complete Synthesis
Problem 7 Solution: End Behavior & Multiplicities P.2G, P.2H
Degree: \( 2 + 1 + 3 = 6 \) (Even)
Leading Term: \( -2 \cdot x^2 \cdot x \cdot x^3 = \mathbf{-2x^6} \) (\( a_n < 0 \))
Limit Notation:
\( \lim_{x \to \infty} f(x) = -\infty \), \( \lim_{x \to -\infty} f(x) = -\infty \)
y-intercept: \( f(0) = -2(3)^2(-1)(-4)^3 = -1152 \)
Zero Analysis Table:
Problem 8 Solution: Function Operations & Composition P.2B
(a) Difference Quotient:
\( f(x+h) = (x+h)^2 - 4(x+h) + 3 \)
\( f(x+h) - f(x) = 2xh + h^2 - 4h \)
\( \frac{f(x+h)-f(x)}{h} = \mathbf{2x + h - 4} \)
(b) Composite Polynomial:
\( (f \circ g)(x) = f(2x - 5) \)
\( = (4x^2 - 20x + 25) - 8x + 20 + 3 \)
\( (f \circ g)(x) = \mathbf{4x^2 - 28x + 48} \)
⚠️ Diagnostic Error: For (a), emphasize that all terms lacking \( h \) must cancel in \( f(x+h) - f(x) \). If constants or \( x^2 \) survive, a distribution sign error occurred.
Tiered Intervention Decision Matrix:
Tier 1 (Mastery 85%+)
Ready for Rational Functions and Asymptote Analysis. Provide difference quotient limit extensions.
Tier 2 (Review 65-84%)
Re-engage with Slide Deck Slides 2 & 5. Practice 3 targeted problems on composition expansion and domain chains.
Tier 3 (Intensive <65%)
1-on-1 guided modeling with color-coded algebraic tiles and function machines prior to re-assessment.
TEKS Intervention: Graph Attributes & Rubric Page 3 of 3 Teacher Solutions Key
Polynomial Recovery Quiz Precalculus Mastery Assessment
Polynomial & Operations Quiz
TEKS P.2B, P.2C, P.2D, P.2E, P.2F, P.2G, P.2H
Name:
Date:
Period: Score: / 100
1. Composition & Restricted Domain (15 pts):
TEKS P.2B
Let \( f(x) = \frac{4}{x - 3} \) and \( g(x) = \sqrt{x + 5} \). Determine \( (f \circ g)(x) \) and find its complete domain in interval notation.
(a) Simplified Composite Function \( (f \circ g)(x) \):
(b) Domain Determination (Interval Notation):
2. Function Operations: Difference Quotient (15 pts):
TEKS P.2B
Given \( f(x) = 2x^2 - 5x + 1 \), compute and simplify the difference quotient \( \frac{f(x+h) - f(x)}{h} \) where \( h \ne 0 \).
Show algebraic expansion, subtraction of \( f(x) \), and final division by \( h \):
3. Invertibility & Algebraic Derivation (15 pts):
TEKS P.2C
Find the inverse function \( f^{-1}(x) \) for the rational function \( f(x) = \frac{3x - 2}{x + 4} \).
Switch variables, solve for \( y \), and write final inverse notation \( f^{-1}(x) \):
Part I: Function Operations & Inverses Page 1 of 2 Precalculus Remediation Quiz
Part II: Polynomial Analysis
Division, Factoring & Graph Signatures
55 Points Remaining
4. Synthetic Division with Missing Degree Terms (15 pts):
TEKS P.2D, P.2E
Divide \( P(x) = 2x^4 - 5x^2 + 4x - 6 \) by \( (x + 2) \). Identify quotient \( Q(x) \), remainder \( R \), and evaluate \( P(-2) \).
Synthetic Division Setup & Table: Quotient: \( Q(x) \) | Remainder: \( R \) | \( P(-2) = \)
5. Complete Factorization & Rational Roots (20 pts):
TEKS P.2D, P.2F
Given that \( x = 2 \) is a known root of \( P(x) = x^3 - 3x^2 - 10x + 24 \), use synthetic division to depress the polynomial, factor completely into linear factors, and list all real rational roots.
Depressed Quadratic Quotient:
Complete Linear Factors & All Real Roots:
6. End Behavior & Axis Intercept Multiplicities (20 pts):
TEKS P.2F, P.2G, P.2H
Given \( f(x) = -(x + 4)(x - 1)^2(x - 3) \):
Leading Term: Degree = _____ , Leading Coeff = _____
Formal End Behavior:
\( \lim_{x \to \infty} f(x) = \) _________
\( \lim_{x \to -\infty} f(x) = \) _________
y-intercept: \( (0, \text{\_\_\_\_\_}) \)
Zero, Multiplicity & Axis Behavior Table:
ZeroMultiplicityAxis Behavior
\( -4 \)
\( 1 \)
\( 3 \)
Part II: Polynomial Structure & Graphs Page 2 of 2 Precalculus Remediation Quiz
Polynomial Quiz Key Teacher Solutions Key 100-Point Assessment
Polynomial Recovery Quiz Key
Part I: Function Operations, Difference Quotients & Inverses (45 pts)
Scoring Standards:
Full Algebraic Work Required
Question 1 Solution: Composition & Domain (15 pts) P.2B
(a) Composition (7 pts):
\( (f \circ g)(x) = f(\sqrt{x+5}) = \frac{4}{\sqrt{x+5} - 3} \)
Award 4 pts for setup, 3 pts for simplified single fraction.
(b) Domain (8 pts):
1. Inner radical: \( x + 5 \ge 0 \implies x \ge -5 \) (4 pts)
Domain: \( [-5, 4) \cup (4, \infty) \) (2 pts)
⚠️ Common Error: Students often forget to exclude \( x = 4 \) or write \( x > -5 \) instead of \( x \ge -5 \). Deduct 2 pts for wrong interval bracket.
Question 2 Solution: Difference Quotient (15 pts) P.2B
Step-by-Step Scoring Breakdown:
1. Expand \( f(x+h) = 2(x+h)^2 - 5(x+h) + 1 = 2x^2 + 4xh + 2h^2 - 5x - 5h + 1 \) (5 pts)
2. Subtract \( f(x) \): \( (2x^2 + 4xh + 2h^2 - 5x - 5h + 1) - (2x^2 - 5x + 1) = 4xh + 2h^2 - 5h \) (5 pts)
3. Divide by \( h \): \( \frac{h(4x + 2h - 5)}{h} = \mathbf{4x + 2h - 5} \) (5 pts)
⚠️ Error Check: If any non-\( h \) terms remain in the numerator before dividing, student dropped a negative sign when subtracting \( f(x) \).
Question 3 Solution: Inverse Function Derivation (15 pts) P.2C
1. Switch variables: \( x = \frac{3y - 2}{y + 4} \) (3 pts)
Quiz Key: Part I (45 pts) Page 1 of 2 Teacher Intervention Guide
Teacher Solutions Key 100-Point Assessment
Polynomial Structure & Graph Key
Part II: Synthetic Division, Factoring & Graph Signatures (55 pts)
Standards Assessed:
P.2D, P.2E, P.2F, P.2G, P.2H
Question 4 Solution: Synthetic Division (15 pts) P.2E
Divide \( 2x^4 + 0x^3 - 5x^2 + 4x - 6 \) by \( (x + 2) \implies c = -2 \):
-2 | 2 0 -5 4 -6 (3 pts for 0x³ placeholder)
-4 8 -6 4
2 -4 3 -2 | -2
Quotient \( Q(x) \): \( 2x^3 - 4x^2 + 3x - 2 \) (5 pts)
Remainder \( R \): \( -2 \) (4 pts)
\( P(-2) \): \( -2 \) (3 pts)
Question 5 Solution: Factoring & Rational Roots (20 pts) P.2D, P.2F
Synthetic Division & Quotient (7 pts):
2 | 1 -3 -10 24
2 -2 -24
1 -1 -12 | 0
\( Q(x) = x^2 - x - 12 \)
Complete Factors & Roots (13 pts):
Factor \( Q(x) = (x - 4)(x + 3) \)
Real Roots: \( \{-3, 2, 4\} \) (6 pts)
Question 6 Solution: Graphs, Limits & Multiplicities (20 pts) P.2G, P.2H
Degree & Leading Term (4 pts):
Degree = 4 (Even), Leading Coeff = -1, Leading Term = -x⁴
Limit Notation (6 pts):
\( \lim_{x \to \infty} f(x) = -\infty, \quad \lim_{x \to -\infty} f(x) = -\infty \)
y-intercept (4 pts): \( f(0) = -(4)(-1)^2(-3) = \mathbf{12} \implies (0, 12) \)
Intercept Table (6 pts):
Mastery (90-100%)
Full algebraic justification
Approaching (75-89%)
Minor arithmetic/sign slips
Intervention (<75%)
Reteach synthetic setup/domain
Quiz Key: Part II (55 pts) | Total: 100 pts Page 2 of 2 Teacher Intervention Guide
