LessonMathematics · Polynomials

Polynomials

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Lesson overview

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This lesson brings together the PDF, study notes and chapter flow so you can revise Polynomials without losing the thread.

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Definitions and operations

Polynomial

P(x) = aₙxⁿ + … + a₀ with aₙ ≠ 0. The degree is n.

P(x) = anx^n + \ldots + a1x + a0

Addition and subtraction

Add/subtract coefficients of the same degree.

Multiplication

deg(P·Q) = deg(P) + deg(Q). Expand term by term.

Division and roots

Euclidean division of polynomials

P = Q·D + R with deg(R) < deg(D). Q is the quotient, R the remainder.

P = Q \cdot D + R, deg(R) < deg(D)

Remainder theorem

The remainder of dividing P by (x − a) is P(a).

P(x) ÷ (x-a) \Longrightarrow reste = P(a)

Factor theorem

a is a root of P ⟺ (x − a) divides P(x), i.e. P(a) = 0.

P(a) = 0 \Longleftrightarrow P(x) = (x-a) \cdot Q(x)

Notable identities

(a+b)²

(a+b)² = a² + 2ab + b²

(a+b)^{2} = a^{2} + 2ab + b^{2}

(a−b)²

(a−b)² = a² − 2ab + b²

(a-b)^{2} = a^{2} - 2ab + b^{2}

a²−b² (difference of squares)

a² − b² = (a−b)(a+b)

a^{2} - b^{2} = (a-b)(a+b)

(a+b)³

(a+b)³ = a³ + 3a²b + 3ab² + b³

(a+b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}

Essential references (Morocco Common Core Science)

Problem-solving method

Identify data, set formulas, solve step by step, then verify sign, unit, and coherence.

Common mistakes

Forgetting domain constraints, mixing equality/equivalence, and losing solutions during transformations.

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