Lesson Notes By Weeks and Term v4 - SHS 2

Lesson Video

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Performance objectives

• Define polynomial functions, factors, and zeros.
• Find the factors and zeros of a polynomial.
• Understand the connection between zeros and a polynomial's graph.
• Relate polynomial functions to real-life situations.

Lesson summary

Polynomials are not just abstract mathematical expressions; they are powerful tools used to model real-world situations. In Ghana, an economist might use a polynomial to model the profit of a small business based on its production levels. An engineer might use one to design the curve of a new road or the arch of a bridge. To understand these models, we need to find their "zeros" – the points where the model equals zero. For a business, this could be the break-even point where there is no profit and no loss. This lesson will equip you with the algebraic tools to find these crucial points.

Lesson notes

Concept 1: The Language of Polynomials - Factors, Roots, Zeros, and Solutions

These terms are often used interchangeably, but they have subtle differences in meaning. Let's clarify them with a familiar quadratic example.

Consider the polynomial function: P(x) = x² + 2x - 15 Factors: These are the expressions that multiply together to give the polynomial. By factoring the quadratic, we get: `P(x) = (x + 5)(x - 3)` So, `(x + 5)` and `(x - 3)` are the factors of `x² + 2x - 15`. Solutions / Roots: When we set the polynomial equal to zero to form an equation, the values of `x` that make the equation true are the solutions or roots. `x² + 2x - 15 = 0` `(x + 5)(x - 3) = 0` This gives us `x + 5 = 0` or `x - 3 = 0`. Therefore, x = -5 and x = 3 are the solutions or roots of the equation. Zeros: The zeros of a polynomial *function* `P(x)` are the input values of `x` for which the output `P(x)` is zero. They are numerically the same as the roots. We say the zeros of `P(x)` are -5 and 3. x-intercepts: When we graph the function `y = P(x)`, the points where the graph crosses the x-axis are the x-intercepts. At these points, the y-coordinate is 0. For our example, the x-intercepts are the points (-5, 0) and (3, 0).

Summary Table:

Evaluation guide

• Find the factors and zeros of a polynomial function using conventional and personal strategy.
• Collaborative Learning, Experiential Learning, Problem -based Learning, Project -Based
• conceptual
• understanding Activity : Zeros of a Polynomial function.