This course takes you through polynomials in the order you need them. You start with the vocabulary: terms, monomials, degree and the standard form, and the names for linear, quadratic, cubic and bi-quadratic polynomials. Then you find values, zeros and roots, solving quadratics by splitting the middle term and by the formula. Next comes division: long division and the division algorithm, the remainder theorem and the factor theorem, with many problems on finding unknown coefficients. Finally you use the factor theorem to factorise cubics and quartics, learn synthetic division as a quick shortcut, and meet the standard factors of xⁿ + aⁿ and xⁿ − aⁿ.
Middle Years
Polynomials and Their Properties
Polynomials from the first definitions to factorisation: terms and degree, zeros and roots, the quadratic formula, long and synthetic division, the remainder and factor theorems, and the factors of xⁿ ± aⁿ.
- 11 lessons
Overview
Lessons
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Terms, monomials and polynomials
What a term is, which terms are monomials, how binomials, trinomials and polynomials are named, and how to find the degree of a monomial and of a polynomial.
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Standard form and types of polynomials
The standard form of a polynomial in x, leading and constant terms, linear, quadratic, cubic and bi-quadratic polynomials, the constant and zero polynomials, and multinomials.
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Value, zeros and roots of a polynomial
Equations by degree, the value p(α) of a polynomial, zeros of a polynomial, roots of an equation, and solving quadratic equations by splitting the middle term.
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The quadratic formula and the discriminant
Proving the formula x = (−b ± √(b² − 4ac)) ÷ 2a by completing the square, the discriminant Δ = b² − 4ac, and using the formula to solve quadratic equations.
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Long division and the division algorithm
Dividing one polynomial by another step by step, finding the quotient and remainder, and checking the answer with f(x) = g(x)q(x) + r(x).
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The remainder theorem
Finding the remainder on division by x − a, ax − b or ax + b without dividing, and using remainders to find unknown coefficients.
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The factor theorem
Testing factors with f(a) = 0, finding unknown coefficients from given factors, proving divisibility without dividing, and what to add or subtract to make a polynomial exactly divisible.
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Factorising quadratics through their zeros
Splitting the middle term, including surd coefficients, and factorising any quadratic as a(x − α)(x − β) once its zeros are found by the formula.
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Factorising cubics and quartics
Using a given factor or a factor found by trial with the factor theorem to factorise cubic and bi-quadratic polynomials completely.
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Synthetic division
A quick table method for dividing a polynomial by x − a, used to find quotients and remainders and to factorise cubics and quartics.
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Factors of xⁿ + aⁿ and xⁿ − aⁿ
When x + a and x − a divide xⁿ ± aⁿ, the expansions of xⁿ − aⁿ and xⁿ + aⁿ, and remainders of large powers such as 13¹⁴¹ + 11¹⁴¹ divided by 24.
Practice papers
Ten question papers that follow the lessons in order. Each has 20 questions for 50 marks (1 hour 30 minutes), with a key sheet and worked solutions.
MCQ tests
Ten tests of 25 multiple-choice questions (30 minutes each). Answer them on the page and check your score, or download the PDF with its key and solutions.
Class notes
Polynomials and Their Properties
Typed from the teacher's handwritten class slides, 27 pages.
Download the notes (PDF, 1 MB)