Polynomial Operations Guided Notes
Algebra 1 • Unit: Polynomials
Name: Date: Period:
Adding Polynomials
| Guided Notes & Practice
Combine like terms • We are simplifying, not solving
Teacher Model Simplify: \((2a^3 + 6a^2 - 3) + (5a^3 - 8a^2 + 5)\)
Method A: Horizontal Grouping
\(= (2a^3 + 5a^3) + (6a^2 - 8a^2) + (-3 + 5)\)
\(= 7a^3 - 2a^2 + 2\)
Method B: Vertical Alignment Stack matching columns:
\(2a^3 + 6a^2 - 3\)
\(+\; 5a^3 - 8a^2 + 5\)
\(7a^3 - 2a^2 + 2\)
Guided Practice • We Do Simplify: \((3x^2 - 5x + 8) + (2x^2 + 7x - 4)\)
Step 1 (Group like terms): \((\;\)_____ \(x^2 + 2x^2) \;+\; (-5x \;+\) _____ \() \;+\; (8 \;+\) _____ \()\)
Step 2 (Combine): \((\;\)_____ \()x^2 \;+\; (\;\)_____ \()x \;+\; (\;\)_____ \()\)
Final Answer:
Independent Practice • You Try Follow the guided steps for each problem
Problem 1: \((4x^3 - 6x^2 + 2x + 7) + (5x^2 - 2x)\) Notice missing \(x^3\) in 2nd polynomial
Step 1: Group like terms by degree or align vertically
Step 2: Add coefficients (keep exponents)
Step 3: Final Answer (Standard Form)
Problem 2: \((7y^2 - 4y + 9) + (3y^2 + 8y - 12)\) Practice
Step 1: Group like terms by degree or align vertically
Step 2: Add coefficients (keep exponents)
Step 3: Final Answer (Standard Form)
Page 1 of 2 • Adding Polynomials
Algebra 1 • Unit: Polynomials
Name: Date: Period:
Subtracting Polynomials
| Guided Notes & Practice
Distribute subtraction → flip signs in 2nd parenthesis → combine
Teacher Model Simplify: \((2a^3 + 6a^2 - 3) - (5a^3 - 8a^2 + 5)\)
Step 1: Distribute \(-1\)
\(2a^3 + 6a^2 - 3\)
\(- 5a^3 + 8a^2 - 5\)
Step 2: Group Terms
\((2a^3 - 5a^3)\)
\(+ (6a^2 + 8a^2) + (-3 - 5)\)
Step 3: Combine
\(= -3a^3 + 14a^2 - 8\)
Guided Practice • We Do Simplify: \((6x^2 + 4x - 9) - (2x^2 - 7x + 3)\)
Step 1 (Distribute \(-1\)): \(6x^2 + 4x - 9 \;\mathbf{-}\;\) _____ \(x^2 \;\mathbf{+}\;\) _____ \(x \;\mathbf{-}\;\) _____
Step 2 (Group & Combine): \((\;\)_____ \()x^2 \;+\; (\;\)_____ \()x \;+\; (\;\)_____ \()\)
Final Answer:
Independent Practice • You Try Follow the guided steps for each problem
Problem 1: \((5x^2 - 3x + 8) - (2x^2 - 7x - 11)\) Watch double negatives: \(-(-7x)\) and \(-(-11)\)
Step 1: Distribute the negative sign (flip signs of 2nd polynomial) and rewrite
Step 2: Group like terms or align vertically
Step 3: Final Answer (Combine)
Problem 2: \((8x^2 - 3x + 5) - (4x^2 + 6x - 7)\) Practice
Step 1: Distribute the negative sign (flip signs of 2nd polynomial) and rewrite
Step 2: Group like terms or align vertically
Step 3: Final Answer (Combine)
Page 2 of 2 • Subtracting Polynomials
Polynomial Operations Answer Key
Algebra 1 • Unit: Polynomials
Teacher Answer Key | Page 1 Solutions
Adding Polynomials • Solutions
| Guided Notes & Practice
Key Rule: Combine coefficients • Exponents stay unchanged!
Teacher Model Simplify: \((2a^3 + 6a^2 - 3) + (5a^3 - 8a^2 + 5)\)
Method A: Horizontal Grouping
\(= (2a^3 + 5a^3) + (6a^2 - 8a^2) + (-3 + 5)\)
\(= 7a^3 - 2a^2 + 2\)
Method B: Vertical Alignment Stack matching columns:
\(2a^3 + 6a^2 - 3\)
\(+\; 5a^3 - 8a^2 + 5\)
\(7a^3 - 2a^2 + 2\)
Guided Practice • We Do (Completed) Simplify: \((3x^2 - 5x + 8) + (2x^2 + 7x - 4)\)
Step 1 (Group like terms): \((\;\)3\(x^2 + 2x^2) \;+\; (-5x \;+\) 7x\() \;+\; (8 \;+\) -4\()\)
Step 2 (Combine): \((\;\)5\()x^2 \;+\; (\;\)2\()x \;+\; (\;\)4\()\)
Final Answer:
\(5x^2 + 2x + 4\)
Independent Practice Solutions • Detailed Worked Steps •
Problem 1: \((4x^3 - 6x^2 + 2x + 7) + (5x^2 - 2x)\) Notice: \(2x - 2x = 0\) (linear term cancels)
Step 1: Group like terms by degree or align vertically
\(= 4x^3 + (-6x^2 + 5x^2) + (2x - 2x) + 7\)
Vertical: \((4x^3 - 6x^2 + 2x + 7) + (0x^3 + 5x^2 - 2x + 0)\)
Step 2: Add coefficients (keep exponents)
\(= 4x^3 + (-1x^2) + 0x + 7\)
Step 3: Final Answer (Standard Form)
\(4x^3 - x^2 + 7\)
Problem 2: \((7y^2 - 4y + 9) + (3y^2 + 8y - 12)\) Standard trinomial addition
Step 1: Group like terms by degree or align vertically
\(= (7y^2 + 3y^2) + (-4y + 8y) + (9 - 12)\)
Vertical: Align \(y^2\) column \((7+3)\), \(y\) column \((-4+8)\), constant column \((9-12)\)
Step 2: Add coefficients (keep exponents)
\(= 10y^2 + 4y + (-3)\)
Step 3: Final Answer (Standard Form)
\(10y^2 + 4y - 3\)
Page 1 of 2 Solutions • Adding Polynomials • Teacher Answer Key
Algebra 1 • Unit: Polynomials
Teacher Answer Key | Page 2 Solutions
Subtracting Polynomials • Solutions
| Guided Notes & Practice
Key Rule: Distribute \(-1\) to EVERY term • Then combine!
Teacher Model Simplify: \((2a^3 + 6a^2 - 3) - (5a^3 - 8a^2 + 5)\)
Step 1: Distribute \(-1\)
\(2a^3 + 6a^2 - 3\)
\(- 5a^3 + 8a^2 - 5\)
Step 2: Group Terms
