Polynomial Operations

Multiplying and dividing polynomials extends the arithmetic you already know from numbers to letters. The product of two linear terms splits a rectangle into four pieces, while synthetic division gives the quotient and remainder fast. Just like 17 = 5 × 3 + 2, every polynomial follows f(x) = divisor × quotient + remainder. Drag the sliders for b and the constant term to reshape the area model and the division table.

Multiplying Polynomials — Area Model

A product of two polynomials is the area of a rectangle. Expanding (x + a)(x + b) is the sum of four sub-rectangles.

value of b4

💡 Why the Area Model Works

①x² is the big square, (a+b)x are two rectangles, ab is the small one

②Sum of all four = the expansion

③This is the foundation of multiplication formulas

Polynomial Division — Synthetic

When dividing by (x − a), synthetic division gives the quotient and remainder quickly.

constant a₀6

🔑 How Synthetic Division Works

①Write a on the left when dividing by (x − a)

②Drop the leading coefficient

③Multiply that by a, add to the next coefficient

④The last entry is the remainder

Division Identity

Polynomial Division Identity

f(x) = (x − a) · Q(x) + R

dividend = divisor × quotient + remainder

Identity Property

(degree n) ÷ (degree 1) → quotient degree n−1, remainder constant

Each division reduces the degree by 1

📐 Compare with Integer Division

①17 ÷ 5 = 3 … 2 → 17 = 5 × 3 + 2

②Polynomials follow the same shape: f(x) = divisor × quotient + remainder

③Remainder degree < divisor degree (always!)

Multiplication Formulas

Perfect Square

(a ± b)² = a² ± 2ab + b²

Expansion of a square

Sum × Difference

(a + b)(a − b) = a² − b²

Difference of squares

Cube Expansion

(a + b)³ = a³ + 3a²b + 3ab² + b³

n = 3 case of the binomial theorem

Work It Out

Example 1

Expand (x + 3)².

Use the perfect-square formula (a + b)² = a² + 2ab + b².

(x + 3)² = x² + 2·x·3 + 3²

Simplify.

x² + 6x + 9

▸ x² + 6x + 9

Memorizing the product formulas speeds up expansion.

Example 2

If x + y = 5 and xy = 3, find x² + y².

Use the identity x² + y² = (x + y)² − 2xy.

x² + y² = (x + y)² − 2xy

Substitute the values.

= 5² − 2·3 = 25 − 6 = 19

▸ x² + y² = 19

Given the sum and product, get a symmetric expression via the identities.

Wrap-up

Core Division Identity

f(x) = (divisor) × Q(x) + R

Backbone of polynomial division

Grade-10 school exam type

If x + y = 4 and xy = 2, what is x³ + y³?

①28

②34

③40

④46

⑤52

▸ ③ 40

Use the identity x³ + y³ = (x + y)³ − 3xy(x + y).

x³ + y³ = (x + y)³ − 3xy(x + y)

Substitute the values.

= 4³ − 3·2·4 = 64 − 24 = 40

🎯 Exam Points

①Multiplication formulas ↔ factoring — master both directions

②Synthetic division feeds directly into the Remainder Theorem

③Plug a number into the identity to find the remainder

④Cube expansion extends to the binomial theorem

⑤Degree: dividend = quotient + divisor

Remainder & Factor Theorem

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