Lesson Notes By Weeks and Term v4 - SHS 1

Lesson Video

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Performance objectives

• Define and explain the Remainder and Factor Theorems.
• Apply the Remainder Theorem to find the remainder of a polynomial division.
• Use the Factor Theorem to determine if an expression is a factor of a polynomial.
• Solve problems involving polynomials using both theorems.

Lesson summary

This lesson introduces two powerful shortcuts in algebra: the Remainder Theorem and the Factor Theorem. We often work with complex polynomial expressions in fields like engineering, economics, and even in planning agricultural projects. Instead of using the long and sometimes tedious method of polynomial long division, these theorems provide a quick and elegant way to find remainders and determine factors. Understanding these tools is fundamental to solving higher-degree polynomial equations, which are essential for modelling real-world situations, such as calculating the profit of a small business in Madina Market or designing a curved bridge support in Accra.

Lesson notes

A. The Foundation: Polynomial Division

Before we learn the shortcuts, let's remember the basic principle of division. When you divide one polynomial, the *dividend* `P(x)`, by another, the *divisor* `D(x)`, you get a *quotient* `Q(x)` and a *remainder* `R(x)`.

This can be written as: `P(x) = D(x) × Q(x) + R(x)`

For example, if we divide `x³ + 2x² - 5x - 6` by `(x - 2)` using long division, we find that the quotient is `x² + 4x + 3` and the remainder is `0`. So, `x³ + 2x² - 5x - 6 = (x - 2)(x² + 4x + 3) + 0`.

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.