Polynomials and Algebraic Processes

Polynomials are algebraic expressions consisting of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. In the JAMB syllabus, the focus is on expressions up to degree 3 (cubic polynomials).

1. Change of Subject of Formula

This involves rearranging an equation to isolate a specific variable. Key Steps:

1. Eliminate fractions by multiplying by the LCM.
2. Remove brackets and collect terms containing the desired subject on one side.
3. Factorize the subject if it appears in multiple terms.
4. Divide to isolate the subject.

2. Factor and Remainder Theorems

Remainder Theorem: If a polynomial f(x) is divided by (x - a), the remainder is f(a).
Factor Theorem: If f(a) = 0, then (x - a) is a factor of f(x).

3. Factorization of Polynomials

Factorization is the process of breaking down a polynomial into a product of simpler factors. Techniques include:

• Difference of Two Squares: a² - b² = (a - b)(a + b)
• Perfect Squares: a² + 2ab + b² = (a + b)²
• Sum/Difference of Cubes: a³ + b³ = (a + b)(a² - ab + b²) and a³ - b³ = (a - b)(a² + ab + b²)
• Grouping: Used for four-term polynomials like ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y).

4. Simultaneous Equations (Linear and Quadratic)

To solve a system consisting of a linear and a quadratic equation, use the Substitution Method:

1. Express one variable in terms of the other from the linear equation.
2. Substitute this into the quadratic equation.
3. Solve the resulting quadratic equation to find two values.
4. Substitute back to find the corresponding values for the second variable.

5. Graphs of Polynomials

Graphs provide visual representations of polynomial functions. For a cubic graph y = ax³ + bx² + cx + d:

• Intercepts: Where the graph crosses the x-axis (roots) and y-axis.
• Turning Points: Local maximum and minimum values where the gradient is zero.
• Shape: An 'S' shape that goes from negative to positive infinity (or vice versa).