Advanced Polynomial Operations

📐 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

$(4x^2 + 3x - 1) - (2x^2 - x + 5)$ $= 4x^2 + 3x - 1 - 2x^2 + x - 5$ $= 2x^2 + 4x - 6$ ✅

Concept Check 🎯

Add/Subtract Polynomials 🧮

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

1. Coefficient of x?

2. Constant term?

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

1. Constant term?

Concept Check 🔍

Practice

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📊 Multiplying Polynomials

Part 2 of 7 — Multiplying Polynomials

Use FOIL for binomials or distribute each term.

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

Special products:

• $(a+b)^2 = a^2 + 2ab + b^2$
• $(a-b)(a+b) = a^2 - b^2$

Worked Example

$(2x + 3)(x - 4)$ $= 2x^2 - 8x + 3x - 12$ $= 2x^2 - 5x - 12$ ✅

Concept Check 🎯

Multiply 🧮

$(x+2)(x+5) = x^2 + ?x + ?$

1. Coefficient of x?

2. Constant term?

$(x+3)^2 = x^2 + 6x + ?$

1. Constant term?

Concept Check 🔍

Practice

Challenge Question 📋

🔢 Factoring Review

Part 3 of 7 — Factoring Review

Methods

1. GCF: Factor out the greatest common factor
2. Trinomial: $x^2 + bx + c = (x+m)(x+n)$ where $m+n=b$, $mn=c$
3. Difference of squares: $a^2-b^2=(a+b)(a-b)$

Worked Example

$x^2 + 7x + 12 = ?$

Find $m, n$: $m + n = 7$, $mn = 12$ → $m = 3, n = 4$

$(x+3)(x+4)$ ✅

Concept Check 🎯

Factor 🧮

$x^2 + 5x + 6 = (x+?)(x+?)$

1. Smaller number?

2. Larger number?

$x^2 - 9 = (x+?)(x-?)$

1. The number?

Concept Check 🔍

Practice

Challenge Question 📋

📈 Polynomial Division

Part 4 of 7 — Polynomial Division

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

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

Worked Example

$(x^2 + 5x + 6) \div (x + 2)$

$= x + 3$ (since $(x+2)(x+3) = x^2+5x+6$) ✅

Concept Check 🎯

Polynomial Division 🧮

1. $(x^2+3x+2)÷(x+1) = x + ?$

2. $(x^2+5x+6)÷(x+2) = x + ?$

3. $(2x^2+6x)÷(2x) = x + ?$

Concept Check 🔍

Practice

Challenge Question 📋

🧮 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 $(x - c)$ is a factor of $p(x)$.

Worked Example

$p(x) = x^2 - 4x + 3$. Find $p(1)$.

$p(1) = 1 - 4 + 3 = 0$ → $(x-1)$ is a factor! ✅

Concept Check 🎯

Remainder Theorem 🧮

1. $p(x) = x^2 - 4x + 3$. $p(1) = ?$

2. $p(x) = x^2 + x - 6$. $p(2) = ?$

3. $p(x) = x^2 + 1$. $p(1) = ?$

Concept Check 🔍

Practice

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🛠️ 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?

$A = (x+5)(x+2) = x^2 + 7x + 10$ ✅

Concept Check 🎯

Applications 🧮

$(x+3)(x+4) = x^2 + ?x + ?$

1. Coefficient of x?

2. Constant?

$x^2 - 1 = (x+1)(x-?)$

1. The number?

Concept Check 🔍

Practice

Challenge Question 📋

🏆 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

$x^2 - 5x + 6 = (x-2)(x-3)$ Zeros: $x = 2, 3$ ✅

Concept Check 🎯

Review 🧮

$(x+1)(x+6) = x^2 + ?x + ?$

1. Coefficient of x?

2. Constant?

3. $p(x)=x^2-1$. $p(3) = ?$

Concept Check 🔍

Practice

Challenge Question 📋