SHS1 Additional Mathematics · Semester 1 Week 15: Applications of Algebra

Lesson notes

Learning Objectives

Indicator: 1.1.2.LI.14 - Use the remainder and factor theorems to find the factors and remainders of a polynomial of degree not greater than 4.

By the end of the lesson, learners can:

State the Remainder Theorem and use it to find the remainder when a polynomial of degree up to 4 is divided by a linear divisor of the form (x - a).
State the Factor Theorem and use it to test whether a given linear expression (x - a) is a factor of a polynomial of degree up to 4.
Apply the Factor Theorem to find unknown coefficients in a polynomial when information about a factor or remainder is given.
Express a polynomial as a product of its linear factors, given one or more factors, and find the zeros of the polynomial.
Solve contextual problems involving polynomial functions by applying the remainder and factor theorems.

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Curriculum details

Strand: Modelling with Algebra (Strand 1)
Sub-strand: Applications of Algebra (1.2)
Content standard: 1.1.2.CS.1 - Demonstrate knowledge and understanding of applying algebraic processes and reasoning involving sequence, functions, and linear programming. 1.1.2.LO.1 Examine, analyse, determine and predict other terms in a pattern/sequence. 1.1.2.LO.2 Distinguish among various types of relations, find the domain and range of, and evaluate functions. 1.1.2.LO.3 Show that a function is injective (into) and/or surjective (onto). Find the inverse, describe the relationship between two variables and establish composite functions. 1.1.2.LO.4 Graph linear and quadratic functions and determine the intercepts. 1.1.2.LO.5 Find graphical and algebraic solutions to a system of three linear equations in three variables and apply them to solve real life problems. 1.1.2.LO.6 Perform algebraic manipulations on polynomial functions and graph polynomial functions. 1.1.2.LO.7 Find the domain, range, and zero of a rational function and state when it is undefined. 1.1.2.LO.8 Identify and describe the order of a matrix, the identity matrix and the zero matrix; find the determinant and perform basic arithmetic operations on 2 by 2 matrices (addition and subtraction).
Indicator: 1.1.2.LI.14 - Use the remainder and factor theorems to find the factors and remainders of a polynomial of degree not greater than 4. Activity 1: Use the factor and remainder theorems to solve problems of polynomial functions in context.
Suggested placement: Semester 1, Week 15 (Week 15 of the year)
Our suggestion, laid out in curriculum order across three terms of twelve weeks. NaCCA does not fix the week, so follow your school's scheme of learning.
Source document note: The official curriculum document has a fault in this entry, so the curriculum text above is transcribed exactly as printed:

exemplars - p.116: adjacent duplicated CambriaMath characters were collapsed, but this PDF also maps some equation glyphs to the wrong letter or operator; verify every mathematical expression in this row against the rendered page
Curriculum reference: NaCCA curriculum document, p. 116

Exemplars (from the NaCCA curriculum)

Collaborative Learning:
- Learners work in groups/pairs to recall key facts about polynomial functions.
- Learners explore and recognise that generally, for any polynomial function 𝑃(𝑥) division by another polynomial results in:
KLML<N%< QN#&L%<N3 = 𝑞 + KLMLOP3 KLMLOP3
and so; 𝐷 = 𝑄 × 𝐷 + 𝑅
Learners recognise that for any polynomial function 𝑃(𝑥), if (𝑥 - 𝑎) is a divisor, then,
𝑃(𝑥) = 𝑞(𝑥). (𝑥 − 𝑎) + 𝑟; where 𝑞(𝑥) is the quotient and 𝑟 is the remainder.
Learners establish the Remainder theorem and use it to solve related problems.
Remainder Theorem: Given any polynomial functions 𝑃(𝑥), if (𝑥 - 𝑎) is a divisor, then, 𝑃(𝑎) = 𝑞(𝑎). (𝑎 − 𝑎) + 𝑟; and 𝑃(𝑎) = 𝑟
Activity 1: Apply the remainder theorem to solve related problems.
Example: find the remainder when the polynomials 𝑓(𝑥) = 4𝑥 , − 𝑥 " − 9𝑥 ) + 2𝑥 − 5 is divided by (𝑥 - 2)
Learners investigate and establish the Factor Theorem and use it to solve related problems.
Factor Theorem: If (𝑥 - 𝑎) is a factor of a polynomial 𝑃(𝑥), then 𝑃(𝑎) = 0.
Activity 1: Apply the Factor theorem to solve related problems.
Example: When 𝑃(𝑥) = 𝑘 " − 3𝑥 ) + 8𝑥 + 36 is divided by (𝑥 + 2); it leaves no remainder.
- Find the value of 𝑘 express 𝑃(𝑥)= (𝑥 + 2)(𝑎 ) + 𝑏 + 𝑐), where 𝑎, 𝑏 𝑎 𝑐 are constants, and find the zeros.
Teaching and Learning Resources:
- Cut out shapes of different colours and orientation
- Maths Technology learning apps, tools and devices
- Chalkboard illustrations
- Worksheets
- Cut-out geometrical shapes of different colours and orientation
Assessment (1.1.2.AS.14). The document marks these depth-of-knowledge levels for this indicator: Level 3 Strategic reasoning.