Advanced Polynomial Operations - Complete Interactive Lesson

Part 1: Adding & Subtracting Polynomials

📐 Adding & Subtracting Polynomials

Part 1 of 7 — Adding & Subtracting Polynomials

Combine like terms (same variable and exponent).

$(3x^2 + 2x) + (5x^2 - x) = 8x^2 + x$

For subtraction: distribute the negative sign first.

Worked Example

$(4x^2 + 3x - 1) - (2x^2 - x + 5)$ ✅

Concept Check 🎯

Add/Subtract Polynomials 🧮

$(3x + 2) + (5x + 1)$

Coefficient of x?
Constant term?

$(2x^2 - 1) + (x^2 + 3)$

Concept Check 🔍

Practice

#	Operation	Result
1	$(3x+2)+(5x+1)$	$8x+3$

Challenge Question 📋

Part 2: Multiplying Polynomials

📊 Multiplying Polynomials

Part 2 of 7 — Multiplying Polynomials

Use FOIL for binomials or distribute each term.

$(a+b)(c+d) = ac + ad + bc + bd$

Part 3: Factoring Review

🔢 Factoring Review

Part 3 of 7 — Factoring Review

Methods

GCF: Factor out the greatest common factor
Trinomial: $x^2 + bx + c = (x+m)(x+n)$ where ,

Part 4: Polynomial Division

📈 Polynomial Division

Part 4 of 7 — Polynomial Division

Long division or synthetic division (for divisor $x - c$ ).

$\frac{x^2 + 3x + 2}{x + 1} = x + 2$

Part 5: Remainder Theorem

🧮 Remainder Theorem

Part 5 of 7 — Remainder Theorem

When dividing $p(x)$ by $(x - c)$ , the remainder equals $p(c)$ .

If $p(c) = 0$ , then is a of .

Part 6: Problem-Solving Workshop

🛠️ Problem-Solving Workshop

Part 6 of 7 — Problem-Solving Workshop

Solve polynomial problems:

Model with polynomial expressions
Factor to find zeros
Use the remainder theorem to test factors

Worked Example

Rectangle: length $(x+5)$ , width $(x+2)$ . Area?

$A = (x +$ ✅

Part 7: Review & Applications

🏆 Review & Applications

Part 7 of 7 — Review & Applications

Key Concepts

Add/subtract: combine like terms
Multiply: FOIL or distribute
Factor: GCF, trinomial, difference of squares
Divide: long or synthetic division
Remainder theorem: $p(c)$ = remainder when $div (x-c)$

Worked Example

$x^{2} - 5 x +$ Zeros: ✅

#	Product	Result
1	$(x+2)(x+5)$	$x^2+7x+10$
2	$(x+3)^2$	$x^2+6x+9$
3	$(x-4)(x+4)$	$x^2-16$

#	Expression	Factored
1	$x^2+5x+6$	$(x+2)(x+3)$
2	$x^2-9$	$(x+3)(x-3)$
3	$2x^2+4x$	$2x(x+2)$

#	Division	Quotient
1	$(x^2+3x+2)÷(x+1)$	$x+2$
2	$(x^2+5x+6)÷(x+2)$	$x+3$
3	$(2x^2+6x)÷(2x)$	$x+3$

#	p(x)	c	p(c)
1	$x^2-4x+3$	1	0
2	$x^2+x-6$	2	0
3	$x^2+1$	1	2

#	Problem	Expression
1	Area: $(x+3)(x+4)$	$x^2+7x+12$
2	Volume: $x(x+1)(x+2)$	$x^3+3x^2+2x$
3	Factor $x^2-1$	$(x+1)(x-1)$

#	Type	Problem
1	FOIL	$(x+1)(x+6)$
2	Factor	$x^2-5x+6$
3	Evaluate	$p(x)=x^2-1$ , $p(3)$