Polynomial Operations

Letters like x and y still follow the same rules as plain arithmetic. A monomial is a number times letters like 3x², a polynomial is a sum of them, and like terms with the same variable and degree can be added. A polynomial times a monomial distributes term by term, like splitting a rectangle into areas. Drag the coefficients and the area model to see why it works.

Intuition — Multiplying With Variables

👀 What is a Monomial?

①A product of numbers and letters: 3x², -5xy, 7

②In 3x², the 3 is the coefficient, x² is the variable part, degree is 2

③A polynomial is a sum of monomials: 3x² + 2x - 1

Combining Like Terms

First Coefficient3

Second Coefficient2

Adding Like Terms

axⁿ + bxⁿ = (a + b)xⁿ

Only terms with the same variable and degree can have their coefficients added

Polynomial × Monomial — Area Model

a (coefficient of x)3

b (constant)2

Distributive Law

a(b + c) = ab + ac

Multiply the outside term into each inside term

Monomial × ÷

Monomial Multiplication

(coeffs × coeffs) × (variables via exponent laws)

3x² × 2x³ = 6x⁵ → coeffs: 3×2=6, exponents: 2+3=5

Monomial Division

6x⁵2x² = 3x³

Divide coefficients; subtract exponents of the same variable

Work It Out

Example 1

Compute 3x² × 4x³.

Multiply coefficients together and like variables together.

= (3 × 4) × (x² × x³)

Apply the laws of exponents.

= 12x⁵

▸ 12x⁵

For a product of monomials, group coefficients and like variables.

Example 2

Expand 2x(3x − 5).

Distribute over each term.

= 2x × 3x − 2x × 5

Simplify.

= 6x² − 10x

▸ 6x² − 10x

(monomial) × (polynomial) expands by the distributive law.

Exam Wrap-up

Core Algebra Operations

Like terms + distributive law + exponent laws

Master these three to solve everything

Grade-8 school exam type

Compute 12a⁵ b³ ÷ 3a² b.

①4a³ b²

②4a³ b³

③9a³ b²

④4a⁷ b⁴

⑤36a⁷ b⁴

▸ ① 4a³ b²

Divide the coefficients and subtract exponents of like variables.

= (12 ÷ 3) × a⁵⁻² × b³⁻¹

Simplify.

= 4a³ b²

🎯 Exam Points

①Like terms must match variable AND degree exactly (x² ≠ x!)

②Distributive law: keep track of signs in front of parentheses

③Monomial product: multiply coeffs, add exponents

④Monomial division: divide coeffs, subtract exponents

⑤Sign trap: -2(3x - 4) = -6x + 8 (the second sign flips)

← Previous

Laws of Exponents

Linear Inequalities

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