grade 9 Polynomial Multiplication & Formulas

Polynomial Multiplication & Formulas

A product that looks messy settles once you draw it as area. Split a square of side a+b into four pieces and you get a², ab, ab, and b², so (a+b)² = a² + 2ab + b² sits on the figure. Change a sign and you have (a−b)²; sum times difference (a+b)(a−b) drops the middle and leaves a² − b². Sliders a and b move those pieces. 101² and 51×49 are fixed examples below, not the slider values.

(a+b)² — Area Model

📐 Idea of the Area Model

①A square with side a+b has area (a+b)²

②Cut it into four pieces: a² + ab + ab + b²

③Therefore (a+b)² = a² + 2ab + b²

④(a+b)² ≠ a²+b². Do not drop the middle 2ab

Perfect-Square Identities

Sum Squared

(a+b)² = a² + 2ab + b²

Square of a sum = each squared + twice product

Difference Squared

(a−b)² = a² − 2ab + b²

Same idea, only the middle sign flips

💡 Sign Tip

①(a+b)²: middle = +2ab

②(a−b)²: middle = −2ab

③End terms a², b² are always positive (they are squares!)

④(x+a)(x+b) = x²+(a+b)x+ab is also an expansion. The reverse of a difference of squares is the next chapter, factoring

(a+b)(a−b) — Difference of Squares

Sum × Difference

(a+b)(a−b) = a² − b²

Sum times difference = difference of squares

Applying the Formulas

Numerical Trick

101² = (100+1)² = 10000+200+1 = 10201

Multiplication formulas turn ugly numbers easy

🧮 Examples

①99² = (100−1)² = 10000−200+1 = 9801

②51×49 = (50+1)(50−1) = 2500−1 = 2499

③(√3+1)² = 3+2√3+1 = 4+2√3

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

Do not drop the middle term 2ab of a perfect square.

Example 2

Expand (x + 2)(x − 5).

Use (x + a)(x + b) = x² + (a + b)x + ab with a = 2, b = −5.

x² + (2 + (−5))x + 2·(−5)

Simplify.

= x² − 3x − 10

▸ x² − 3x − 10

The linear coefficient is the sum, the constant the product, of the two numbers.

Exam Wrap-up

Three Identities

(a±b)² = a²±2ab+b², (a+b)(a−b) = a²−b²

Foundations of factoring and quadratics

Grade-9 school exam type

Expand (x + 4)(x − 4).

①x² − 8

②x² + 16

③x² − 16

④x² − 8x − 16

⑤x² − 16x

▸ ③ x² − 16

Use the sum-and-difference product (a + b)(a − b) = a² − b².

(x + 4)(x − 4) = x² − 4²

Compute.

= x² − 16

🎯 Exam Points

①(a+b)² ≠ a²+b² — never drop 2ab

②Difference of squares appears more in reverse (factoring)

③Same formulas work with radicals/fractions

④Memorize (x+a)(x+b) = x²+(a+b)x+ab

⑤Always combine like terms after expansion

← Previous

Operations with Radicals

Factorization

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