Algebra

A polynomial is an expression made of:

terms (like $3x^2$ , , )
added or subtracted together
with variables raised to whole number exponents (0, 1, 2, 3, …)

Examples of polynomials:

✅
✅ $x^2 – 4x + 1$
✅ $7$ (a constant is still a polynomial)

NOT polynomials:

❌ (because $x$ is in the denominator → exponent would be -1)
❌ (because exponent is $1/2$ )
❌ $x^{-3}$ (negative exponent)

⭐ What does “polynomial operations” mean?

Polynomial operations are the ways we combine polynomials using:

Adding polynomials
Subtracting polynomials
Multiplying polynomials
(sometimes) Dividing polynomials (usually later)

1️⃣ Adding Polynomials

When you add polynomials, you combine like terms.

Example:

$(3x^2 + 2x + 5) + (x^2 – 4x + 1)$

Group like terms:

$3x^2 + x^2 = 4x^2$

✅ Answer:

$4x^2 – 2x + 6$

2️⃣ Subtracting Polynomials

When subtracting, you distribute the negative sign to every term in the second polynomial.

Example:

$(5x^2 + 3x – 1) – (2x^2 – x + 4)$

Distribute the minus:

$5x^2 + 3x – 1 – 2x^2 + x – 4$

Combine like terms:

$5x^2 – 2x^2 = 3x^2$

✅ Answer:

$3x^2 + 4x – 5$

3️⃣ Multiplying Polynomials

This is where things level up.

A) Multiply a monomial times a polynomial (Distributive Property)

Example:

$3x(2x^2 – x + 4)$

Multiply 3x into each term:

$2x^2 = 6x^3$
$=−3x^2$

✅ Answer:

$3x^2 + 12x$

B) Multiply polynomial times polynomial (FOIL / Distribute)

Example:

FOIL:

First: $x⋅x=x^2$
Outer:
Inner:
Last:

Combine like terms:

$x^2 + 7x + 10$

✅ Answer:

$x^2+7x+10$

4️⃣ Dividing Polynomials (Quick Intro)

This is usually later, but the most common type is:

Example:

$6x^2+12x / 6x$

Divide each term:

$6x^2/6x + 12x/6x = x+2$

✅ Answer: