Lesson Notes By Weeks and Term v4 - SHS 1

Lesson Video

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Performance objectives

• Define polynomial functions and their degree.
• State and explain the Remainder and Factor Theorems.
• Use the Remainder Theorem to find the remainder of a polynomial division.
• Use the Factor Theorem to find factors of polynomials.
• Solve real-life problems using polynomial functions.

Lesson summary

This lesson introduces two powerful "shortcuts" in algebra: the Remainder Theorem and the Factor Theorem. In our daily lives, we often want to find out how things are related – for example, how the profit of a small `kelewele` business changes with the price, or how the height of a thrown stone changes over time. These relationships can often be described by mathematical expressions called polynomials. Instead of using the long and sometimes tedious method of long division to analyse these polynomials, the Remainder and Factor Theorems provide a quick and elegant way to find remainders and determine factors.

Lesson notes

A. The Division Algorithm for Polynomials

Let's start with something we already know: simple division with numbers. When we divide 23 by 5, we get 4 with a remainder of 3. We can write this as: `23 = 5 × 4 + 3` In general: `Dividend = Divisor × Quotient + Remainder`

This same rule applies to polynomials. If we have a polynomial `P(x)` and we divide it by another polynomial `D(x)`, we get a quotient `Q(x)` and a remainder `R(x)`.

`P(x) = D(x) × Q(x) + R(x)`

Evaluation guide

• Use the remainder and factor theorems to find the factors and remainders of a
• polynomial of degree not greater than
• Use the factor and remainder theorems to solve problems of polynomial functions in
• context.